7. Max-Min Problems

Exercises

    Find all critical points of each function. Then use the Second Derivative Test to classify each as a local minimum, local maximum or saddle or say the test fails.

  1. f(x,y)=(x+3)2+(y−1)2f(x,y)=(x+3)^2+(y-1)^2

    Hint

    Set the xx-derivative and the yy-derivative equal to zero and solve for xx and yy. fx(x,y)=∂f∂x=0fy(x,y)=∂f∂y=0\begin{aligned} f_x(x,y)&=\dfrac{\partial f}{\partial x}=0 \\ f_y(x,y)&=\dfrac{\partial f}{\partial y}=0 \end{aligned}

    [×]

    Answer

    The critical point, (−3,1)(-3,1), is a local minimum.

    [×]

    Solution

    We compute the partial derivatives, set them equal to 00 and solve for xx and yy: fx=2(x+3)=0⟹x=−3fy=2(y−1)=0⟹y=1\begin{aligned} f_x=2(x+3)=0 \qquad &\Longrightarrow& \qquad x=-3 \\ f_y=2(y-1)=0 \qquad &\Longrightarrow& \quad y=1 \end{aligned} So the critical point is (x,y)=(−3,1)(x,y)=(-3,1). fxx=2fyy=2fxy=0D=fxxfyy−fxy2=4 f_{xx}=2 \qquad f_{yy}=2 \qquad f_{xy}=0 \qquad D=f_{xx}f_{yy}-{f_{xy}}^2=4 Since D=4>0D=4 \gt 0 and fxx=2>0f_{xx}=2 \gt 0 the critical point is a local minimum.

    [×]

    Remark

    This should have been obvious, since the function is an elliptic paraboloid opening upward with vertex at (−3,1)(-3,1).

    [×]
  2. z=4xy+2yz=4xy+2y

    Answer

    The critical point, (x,y)=(−12,0)(x,y)=\left(-\,\dfrac{1}{2},0\right), is a saddle.

    [×]

    Solution

    ∂z∂x=4y=0⟹y=0∂z∂y=4x+2=0⟹x=−12\begin{aligned} \dfrac{\partial z}{\partial x}=4y=0 \qquad &\Longrightarrow& \quad y=0 \\ \dfrac{\partial z}{\partial y}=4x+2=0 \qquad &\Longrightarrow& \quad x=-\,\dfrac{1}{2} \end{aligned} The critical point is (x,y)=(−12,0)(x,y)=\left(-\,\dfrac{1}{2},0\right). fxx=0fyy=0fxy=4D=fxxfyy−fxy2=−16 f_{xx}=0 \qquad f_{yy}=0 \qquad f_{xy}=4 \qquad D=f_{xx}f_{yy}-{f_{xy}}^2=-16 Since D=−16<0D=-16 \lt 0, the critical point is a saddle.

    [×]
  3. g(x,y)=3x2y+2xy2g(x,y)=3x^2y+2xy^2

    Answer

    The only critical point is (0,0)(0,0) for which the test fails.

    [×]

    Solution

    gx=6xy+2y2=2y(3x+y)=0⟹y=0ory=−3xgy=3x2+4xy=x(3x+4y)=0⟹x=0ory=−3x4\begin{aligned} g_x=6xy+2y^2=2y(3x+y)=0 \qquad &\Longrightarrow& \qquad y=0 \qquad \text{or} \qquad y=-3x \\ g_y=3x^2+4xy=x(3x+4y)=0 \qquad &\Longrightarrow& \qquad x=0 \qquad \text{or} \qquad y=-\,\dfrac{3x}{4} \end{aligned} There are four cases. Three lead to the critical point (x,y)=(0,0)(x,y)=(0,0). The fourth y=−3x=−3x4y=-3x=-\,\dfrac{3x}{4} is a contradiction unless x=y=0x=y=0 again. We apply the Second Derivative test: gxx=6ygyy=4xgxy=6x+4y g_{xx}=6y \qquad g_{yy}=4x \qquad g_{xy}=6x+4y So gxx(0,0)=0gyy(0,0)=0gxy(0,0)=0D=0 g_{xx}(0,0)=0 \qquad g_{yy}(0,0)=0 \qquad g_{xy}(0,0)=0 \qquad D=0 Since D=0D=0, the test fails.

    [×]
  4. p(u,v)=3u2−uv2+v3p(u,v)=3u^2-uv^2+v^3

    Answer

    The critical point (u,v)=(272,9)(u,v)=\left(\dfrac{27}{2},9\right) is a local minimum.
    The test fails for the critical point (u,v)=(0,0)(u,v)=(0,0).

    [×]

    Solution

    pu=6u−v2=0(1)pv=−2uv+3v2=v(−2u+3v)=0(2)\begin{aligned} p_u&=6u-v^2=0 \qquad \text{(1)} \\ p_v&=-2uv+3v^2=v(-2u+3v)=0 \qquad \text{(2)} \end{aligned} Equation (2) leads to 22 cases:

    • Case 1: v=0v=0: Then equation (1) says u=0u=0 and gives the critical point (u,v)=(0,0)(u,v)=(0,0).
    • Case 2: v=23uv=\dfrac{2}{3}u: Then equation (1) says 6u−49u2=06u-\,\dfrac{4}{9}u^2=0 or u(6−49u)=0u\left(6-\,\dfrac{4}{9}u\right)=0 which leads to 22 cases:
    • Case 2a: u=0u=0 which repeats the critical point (u,v)=(0,0)(u,v)=(0,0).
    • Case 2b: u=272u=\dfrac{27}{2} and hence v=9v=9 and the critical point (u,v)=(272,9)(u,v)=\left(\dfrac{27}{2},9\right)

    The second derivatives are: puu=6pvv=2u+6vpuv=−2v p_{uu}=6 \qquad p_{vv}=2u+6v \qquad p_{uv}=-2v We make a table to keep track of the values.

    uu vv puup_{uu} pvvp_{vv} puvp_{uv} DD Classification
    00 00 66 00 00 00 test fails
    272\dfrac{27}{2} 99 66 8181 −18-18 162162 minimum
    [×]
  5. f(x,y)=x2+2y2+xy+4x−5yf(x,y)=x^2+2y^2+xy+4x-5y

    Answer

    The critical point, (x,y)=(−3,2)(x,y)=(-3,2), is a local minimum.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=2x+y+4=0(1)fy=x+4y−5=0(2)\begin{aligned} f_x&=2x+y+4&=0 \qquad \text{(1)} \\ f_y&=x+4y-5&=0 \qquad \text{(2)} \end{aligned} Equation (1) minus 22 times equation (2) gives −7y+14=0-7y+14=0. So y=2y=2. Then equation (2) says x=5−4y=−3x=5-4y=-3. This gives the critical point (−3,2)(-3,2). The second derivatives are: fxx=2fyy=4fxy=1D=2⋅4−12=7 f_{xx}=2 \qquad f_{yy}=4 \qquad f_{xy}=1 \qquad D=2\cdot4-1^2=7 Since D>0D \gt 0 and fxx>0f_{xx} \gt 0 the critical point is a local minimum.

    [×]
  6. f(x,y)=2x2y+2xy2−3y2f(x,y)=2x^2y+2xy^2-3y^2

    Answer

    The critical point (u,v)=(2,−4)(u,v)=(2,-4) is a saddle point.
    The test fails for the critical point (u,v)=(0,0)(u,v)=(0,0).

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=4xy+2y2=0(1)fy=2x2+4xy−6y=0(2)\begin{aligned} f_x&=4xy+2y^2&=0 \qquad \text{(1)} \\ f_y&=2x^2+4xy-6y&=0 \qquad \text{(2)} \end{aligned} Equation (1) factors as y(4x+2y)=0y(4x+2y)=0. This gives us two cases:

    • Case 1: y=0y=0: Then equation (2) reduces to 2x2=02x^2=0 or x=0x=0. This gives the critical point (0,0)(0,0).
    • Case 2: y=−2xy=-2x: Then equation (2) reduces to 2x2−8x2+12x=02x^2-8x^2+12x=0 which factors as 6x(−x+2)=06x(-x+2)=0. This gives two cases: x=0x=0 and x=2x=2. These give the the critical points (0,0)(0,0) and (2,−4)(2,-4).

    The second derivatives are: fxx=4yfyy=4x−6fxy=4x+4y f_{xx}=4y \qquad f_{yy}=4x-6 \qquad f_{xy}=4x+4y We make a table to keep track of the values.

    xx yy fxxf_{xx} fyyf_{yy} fxyf_{xy} DD Classification
    00 00 00 −6-6 00 00 test fails
    22 −4-4 −16-16 22 −8-8 −96-96 saddle point
    [×]
  7. f(x,y,z)=2x3+3x2y+3x2z+3x2+y2−z2+2yz−3y+9zf(x,y,z)=2x^3+3x^2y+3x^2z+3x^2+y^2-z^2+2yz-3y+9z

    Answer

    The 33 critical points, (0,−32,3)(−1,−3,3)(53,−173,3) \left(0,-\,\dfrac{3}{2},3\right) \qquad (-1,-3,3) \qquad \left(\dfrac{5}{3},-\,\dfrac{17}{3},3\right) are all saddle points.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=6x2+6xy+6xz+6x=0(1)fy=3x2+2y+2z−3=0(2)fz=3x2+2y−2z+9=0(3)\begin{aligned} f_x&=6x^2+6xy+6xz+6x&=0 \qquad \text{(1)} \\ f_y&=3x^2+2y+2z-3&=0 \qquad \text{(2)} \\ f_z&=3x^2+2y-2z+9&=0 \qquad \text{(3)} \end{aligned} First notice that equation (2) minus equation (3) gives 4z−12=04z-12=0 or z=3z=3. Then equation (2) or (3) reduces to: 3x2+2y+3=0(4) 3x^2+2y+3=0 \qquad \text{(4)} Next notice that equation (1) factors as 6x(x+y+z+1)=06x(x+y+z+1)=0. This gives 22 cases:

    • Case 1: x=0x=0: Then equation (4) reduces to 2y+3=02y+3=0 or y=−32y=-\,\dfrac{3}{2}. This gives the critical point (0,−32,3)\left(0,-\,\dfrac{3}{2},3\right).
    • Case 2: x+y+z+1=0x+y+z+1=0: Using z=3z=3, this says x=−y−4x=-y-4. So equation (4) becomes 3(−y−4)2+2y+3=03(-y-4)^2+2y+3=0 which expands to 3y2+26y+51=0 3y^2+26y+51=0 which has the solutions y=−3y=-3 and y=−173y=-\,\dfrac{17}{3}. This gives two critical points (−1,−3,3)(-1,-3,3) and (53,−173,3)\left(\dfrac{5}{3},-\,\dfrac{17}{3},3\right).

    To apply the Second Derivative Test, we need the Hessian: Hessf=(fxxfxyfxzfyxfyyfyzfzxfzyfzz)=(12x+6y+6z+66x6x6x226x2−2) \text{Hess}f=\begin{pmatrix} f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \\ \end{pmatrix} =\begin{pmatrix} 12x+6y+6z+6 & 6x & 6x \\ 6x & 2 & 2 \\ 6x & 2 & -2 \\ \end{pmatrix} We evaluate at each critical point:

    • At (0,−32,3)\left(0,-\,\dfrac{3}{2},3\right), Hessf=(150002202−2) \text{Hess}f=\begin{pmatrix} 15 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 2 & -2 \\ \end{pmatrix} The Leading Principal Minor Determinants are D1=15D2=30D3=−120 D_1=15\qquad D_2=30 \qquad D_3=-120 They are not all positive and they do not alternate signs and D3≠0D_3\ne0. So this is a saddle point.
    • At (−1,−3,3)(-1,-3,3), Hessf=(−6−6−6−622−62−2) \text{Hess}f=\begin{pmatrix} -6 & -6 & -6 \\ -6 & 2 & 2 \\ -6 & 2 & -2 \\ \end{pmatrix} The Leading Principal Minor Determinants are D1=−6D2=−48D3=192 D_1=-6\qquad D_2=-48 \qquad D_3=192 So this is also a saddle point.
    • At (53,−173,3)\left(\dfrac{5}{3},-\,\dfrac{17}{3},3\right), Hessf=(1010101022102−2) \text{Hess}f=\begin{pmatrix} 10 & 10 & 10 \\ 10 & 2 & 2 \\ 10 & 2 & -2 \\ \end{pmatrix} The Leading Principal Minor Determinants are D1=10D2=−80D3=320 D_1=10\qquad D_2=-80 \qquad D_3=320 Again this is a saddle point.
    [×]
  8. g(x,y)=x3−y3−6xyg(x,y)=x^3-y^3-6xy

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a saddle point.
    The critical point, (x,y)=(−2,2)(x,y)=(-2,2), is a local maximum.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: gx=3x2−6y=0(1)gy=−3y2−6x=0(2)\begin{aligned} g_x&=3x^2-6y&=0 \qquad \text{(1)} \\ g_y&=-3y^2-6x&=0 \qquad \text{(2)} \end{aligned} Solving equation (1) for yy gives the equation y=x22y=\dfrac{x^2}{2}. Plugging this into equation (2) gives: −3(x22)2−6x=0x4+8x=0x(x3+8)=0\begin{aligned} -3\left(\dfrac{x^2}{2}\right)^2-6x&=0 \\ x^4+8x&=0 \\ x(x^3+8)&=0 \end{aligned} So the solutions are (x,y)=(0,0)(x,y)=(0,0) and (x,y)=(−2,2)(x,y)=(-2,2).

    The second derivatives are: gxx=6xgyy=−6ygxy=−6 g_{xx}=6x \qquad g_{yy}=-6y \qquad g_{xy}=-6 We make a table of values:

    xx yy gxxg_{xx} gyyg_{yy} gxyg_{xy} DD Classification
    00 00 00 00 −6-6 −36-36 Saddle Point
    −2-2 22 −12-12 −12-12 −6-6 108108 Local Maximum

    wll 

    [×]
  9. f(x,y)=3x2+6xy−2y3+6y2f(x,y)=3x^2+6xy-2y^3+6y^2

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a local minimum.
    The critical point, (x,y)=(−1,1)(x,y)=(-1,1), is a saddle point.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=6x+6y=0(1)fy=6x−6y2+12y=0(2)\begin{aligned} f_x&=6x+6y&=0 \qquad \text{(1)} \\ f_y&=6x-6y^2+12y&=0 \qquad \text{(2)} \end{aligned} Subtracting equation (2) from equation (1) results in the equation 6y2−6y=06y^2-6y=0 which factors as y(y−1)=0y(y-1)=0. So the solutions are y=0y=0 and y=1y=1.

    • Case 1: y=0y=0: Plugging this into equation (1) yields x=0x=0 and the critical point (0,0)(0,0).
    • Case 2: y=1y=1: Plugging this into equation (1) yields x=−1x=-1 and the critical point (−1,1)(-1,1).

    The second derivatives are: fxx=6fyy=−12y+12fxy=6 f_{xx}=6 \qquad f_{yy}=-12y+12 \qquad f_{xy}=6 We make a table:

    xx yy fxxf_{xx} fyyf_{yy} fxyf_{xy} DD Classification
    00 00 66 1212 66 3636 Local Minimum
    −1-1 11 66 00 66 −36-36 Saddle Point

    wll 

    [×]
  10. g(x,y)=x3+y3−12xyg(x,y)=x^3+y^3-12xy

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a Saddle Point.
    The critical point, (x,y)=(4,4)(x,y)=(4,4), is a Local Minimum.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: gx=3x2−12y=0(1)gy=3y2−12x=0(2)\begin{aligned} g_x&=3x^2-12y&=0 \qquad \text{(1)} \\ g_y&=3y^2-12x&=0 \qquad \text{(2)} \end{aligned} Solving for yy in equation (1) gives y=x24y=\dfrac{x^2}{4}. Plugging this into equation (2) gives: 3(x24)2−12x=0x4−64x=0x(x3−64)=0\begin{aligned} 3\left( \dfrac{x^2}{4} \right)^2 -12x = 0 \\ x^4-64x=0 \\ x(x^3-64) = 0 \end{aligned} So the solutions are (x,y)=(0,0)(x,y)=(0,0) and (x,y)=(4,4)(x,y)=(4,4).
    The second derivatives are: gxx=6xgyy=6ygxy=−12 g_{xx}=6x \qquad g_{yy}=6y \qquad g_{xy}=-12 We make a table of values:

    xx yy gxxg_{xx} gyyg_{yy} gxyg_{xy} DD Classification
    00 00 00 00 −12-12 −144-144 Saddle Point
    44 44 2424 2424 −12-12 432432 Local Minimum

    wll 

    [×]
  11. f(x,y)=x3+y3−3x2+3y2f(x,y)=x^3+y^3-3x^2+3y^2

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a saddle point.
    The critical point, (x,y)=(0,−2)(x,y)=(0,-2), is a local maximum.
    The critical point, (x,y)=(2,0)(x,y)=(2,0), is a local minimum.
    The critical point, (x,y)=(2,−2)(x,y)=(2,-2), is a saddle point.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=3x2−6x=0(1)fy=3y2+6y=0(2)\begin{aligned} f_x&=3x^2-6x&=0 \qquad \text{(1)} \\ f_y&=3y^2+6y&=0 \qquad \text{(2)} \end{aligned} Since each first partial derivatve is a function of only one variable, We can solve them separately.
    Equation (1) factors as 3x(x−2)=03x(x-2)=0. The solutions are x=0,2x=0,2.
    Equation (2) factors as 3y(y+2)=03y(y+2)=0. The solutions are y=0,−2y=0,-2.
    So there are 44 critical points: (0,0)(0,−2)(2,0)(2,−2) (0,0) \qquad (0,-2) \qquad (2,0) \qquad (2,-2)

    The second derivatives are: fxx=6x−6fyy=6y+6fxy=0\begin{aligned} f_{xx}=6x-6 \qquad f_{yy}=6y+6 \qquad f_{xy}=0 \end{aligned} We make a table:

    xx yy fxxf_{xx} fyyf_{yy} fxyf_{xy} DD Classification
    00 00 −6-6 66 00 −36-36 Saddle Point
    00 −2-2 −6-6 −6-6 00 3636 Local Maximum
    22 00 66 66 00 3636 Local Minimum
    22 −2-2 66 −6-6 00 −36-36 Saddle Point

    wll 

    [×]
  12. g(x,y)=x4+y4−4xyg(x,y)=x^4+y^4-4xy

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a saddle point.
    The critical point, (x,y)=(−1,−1)(x,y)=(-1,-1), is a local minimum.
    The critical point, (x,y)=(1,1)(x,y)=(1,1), is a local minimum.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: gx=4x3−4y=0(1)gy=4y3−4x=0(2)\begin{aligned} g_x&=4x^3 - 4y&=0 \qquad \text{(1)} \\ g_y&=4y^3 - 4x&=0 \qquad \text{(2)} \end{aligned} Solving equation (1) for yy yields y=x3y=x^3. Plugging this into equation (2) gives: 4(x3)3−4x=0x(x8−1)=0\begin{aligned} 4\left(x^3\right)^3 - 4x = 0 \\ x(x^8 - 1) = 0 \end{aligned} Solutions are x=0x = 0 and x=±1x = \pm 1. So the 33 critical points are: (0,0)(−1,−1)(1,1) (0,0) \qquad (-1,-1) \qquad (1,1)

    The second derivatives are: gxx=12x2gyy=12y2gxy=−4 g_{xx}=12x^2 \qquad g_{yy}=12y^2 \qquad g_{xy}=-4 We make a table of values:

    xx yy gxxg_{xx} gyyg_{yy} gxyg_{xy} DD Classification
    00 00 00 00 −4-4 −16-16 Saddle Point
    −1-1 −1-1 1212 1212 −4-4 128128 Local Minimum
    11 11 1212 1212 −4-4 128128 Local Minimum

    wll 

    [×]
  13. f(x,y)=x4+y4+4xyf(x,y)=x^4+y^4+4xy

    Answer

    The critical point, (x,y)=(0,0)(x,y)=(0,0), is a saddle point.
    The critical point, (x,y)=(−1,1)(x,y)=(-1,1), is a local minimum.
    The critical point, (x,y)=(1,−1)(x,y)=(1,-1), is a local minimum.

    [×]

    Solution

    To find the critical points, we set the partial derivatives equal to 00 and solve: fx=4x3+4y=0(1)fy=4y3+4x=0(2)\begin{aligned} f_x&=4x^3 + 4y&=0 \qquad \text{(1)} \\ f_y&=4y^3 + 4x&=0 \qquad \text{(2)} \end{aligned} Solving equation (1) for yy yields y=−x3y=-x^3. Plugging this into equation (2) gives: 4(−x3)3+4x=0−4x9+4x=0x(−x8+1)=0\begin{aligned} 4\left(-x^3\right)^3 + 4x = 0 \\ -4x^9+4x = 0 \\ x(-x^8 + 1) = 0 \end{aligned} Solutions are x=0x = 0 and x=±1x = \pm 1. So the 33 critical points are: (0,0)(−1,1)(1,−1) (0,0) \qquad (-1,1) \qquad (1,-1)

    The second derivatives are: fxx=12x2fyy=12y2fxy=4 f_{xx}=12x^2 \qquad f_{yy}=12y^2 \qquad f_{xy}=4 We make a table of values:

    xx yy fxxf_{xx} fyyf_{yy} fxyf_{xy} DD Classification
    00 00 00 00 44 −16-16 Saddle Point
    −1-1 11 1212 1212 44 128128 Local Minimum
    11 −1-1 1212 1212 44 128128 Local Minimum

    wll 

    [×]
  14. Find the dimensions and volume of the largest box with base on the xyxy plane and upper vertices on the paraboloid z=16−2x2−y2z=16-2x^2-y^2.
    Solve by eliminating zz.

    x_box_under_parab

    Hint

    The length, width and height are not just xx, yy and zz.

    [×]

    Answer

    The length, width, height and volume are: L=22W=4H=8V=LWH=642 L=2\sqrt{2} \qquad W=4 \qquad H=8 \qquad V=LWH=64\sqrt{2}

    [×]

    Solution

    The volume of the box is given by V=(2x)(2y)(z)=4xyz V=(2x)(2y)(z)=4xyz where xx, yy and zz are the coordinates of the vertex of the box in the first quadrant. We can eliminate a variable by substituting the constraint equation z=16−2x2−y2z=16-2x^2-y^2: V=4xy(16−2x2−y2)=64xy−8x3y−4xy3 V=4xy(16-2x^2-y^2)=64xy-8x^3y-4xy^3 Next we set the xx and yy partial derivatives equal to zero and solve for xx and yy: fx=64y−24x2y−4y3=4y(16−6x2−y2)=0fy=64x−8x3−12xy2=4x(16−2x2−3y2)=0\begin{aligned} f_x&=64y-24x^2y-4y^3=4y(16-6x^2-y^2)=0 \\ f_y&=64x-8x^3-12xy^2=4x(16-2x^2-3y^2)=0 \end{aligned} If x=0x=0 or y=0y=0, then the volume would be 00 which would not be the maximum. So taking x≠0x\neq 0 and y≠0y\neq 0, we have 6x2+y2=16(1)2x2+3y2=16(2)\begin{aligned} 6x^2+y^2&=16 \qquad \text{(1)} \\ 2x^2+3y^2&=16 \qquad \text{(2)} \end{aligned} We compute 33 times equation (1) minus equation (2): 16x2=32⟹x=2 16x^2=32 \quad \Longrightarrow \qquad x=\sqrt{2} Similarly, 33 times equation (2) minus equation (1) gives: 8y2=32⟹y=2 8y^2=32 \quad \Longrightarrow \qquad y=2 (We ignored the ±\pm because we want (x,y)(x,y) in the first quadrant.) We plug into the parabola: z=16−2x2−y2=16−4−4=8 z=16-2x^2-y^2=16-4-4=8 So the length, width, height and volume are: L=2x=22W=2y=4H=z=8V=LWH=642\begin{aligned} L=2x=2\sqrt{2} \qquad &W=2y=4 \qquad H=z=8 \\ V=&LWH=64\sqrt{2} \end{aligned}

    py,tj 

    [×]
  15. Find the point on the plane 2x−2y−z=182x-2y-z=18 that is closest to the origin. Then find the distance from the point to the origin.
    Solve by eliminating zz.
    Rotate the second plot with your mouse.

    x_dist_pt2origin
    1

    Hint

    It is easier to minimize the square of the distance than the actual distance. The answer will be the same.

    [×]

    Answer

    The closest point is (x,y,z)=(4,−4,−2)(x,y,z)=(4,-4,-2).
    Its distance from the origin is D=6D=6.

    [×]

    Solution

    We need to minimize the distance from the origin to the point (x,y,z)(x,y,z) on the plane D=x2+y2+z2 D=\sqrt{x^2+y^2+z^2} It is easier to minimize the square of the distance, which will give the same solution. So we will minimize the function f=D2=x2+y2+z2 f=D^2=x^2+y^2+z^2 Since the point must lie on the plane, we substitute the equation of the plane z=2x−2y−18z=2x-2y-18 into ff: f=x2+y2+(2x−2y−18)2 f=x^2+y^2+(2x-2y-18)^2 We take the xx and yy derivatives and set them equal to zero: fx=2x+4(2x−2y−18)=10x−8y−72=0fy=2y−4(2x−2y−18)=−8x+10y+72=0\begin{aligned} f_x&=2x+4(2x-2y-18)=10x-8y-72=0 \\ f_y&=2y-4(2x-2y-18)=-8x+10y+72=0 \end{aligned} Equivalently: 5x−4y=36(1)−4x+5y=−36(2)\begin{aligned} 5x-4y&=36 \qquad \text{(1)} \\ -4x+5y&=-36 \qquad \text{(2)} \end{aligned} Multiply the first equation by 44 and the second equation by 55 and add: 20x−16y=144−20x+25y=−1809y=−36⟹y=−4\begin{aligned} 20x-16y&=144 \\ -20x+25y&=-180 \\ 9y&=-36 \qquad \Longrightarrow \qquad y=-4 \end{aligned} Substituting back we have: 5x−4(−4)=36⟹x=4 5x-4(-4)=36 \qquad \Longrightarrow \qquad x=4 z=2x−2y−18=2(4)−2(−4)−18=−2 z=2x-2y-18=2(4)-2(-4)-18=-2 So the closest point is (x,y,z)=(4,−4,−2)(x,y,z)=(4,-4,-2). Its distance from the origin is D=16+16+4=6 D=\sqrt{16+16+4}=6

    py,tj 

    [×]
  16. Find the dimensions and volume of the largest box with base on the xyxy plane and upper vertices on the paraboloid z=16−2x2−y2z=16-2x^2-y^2.
    Solve by using a Lagrange multiplier.

    x_box_under_parab

    Hint

    What are the length, width and height in terms of the point (x,y,z)(x,y,z) on the paraboloid in the first octant.

    [×]

    Answer

    The length, width, height and volume are:
    L=22W=4H=8 L=2\sqrt{2} \qquad W=4 \qquad H=8
    V=642 V=64\sqrt{2}

    [×]

    Solution

    We need to maximize the volume. If we take the point (x,y,z)(x,y,z) in the first octant, the height is H=zH=z, but the length and width need to be doubled: L=2xL=2x and W=2yW=2y. So the volume is V=LWH=4xyzV=LWH=4xyz. This is subject to the constraint that the point (x,y,z)(x,y,z) lies on the paraboloid, g=z+2x2+y2=16g=z+2x^2+y^2=16. The gradients are: ∇⃗V=⟨4yz,4xz,4xy⟩∇⃗g=⟨4x,2y,1⟩\begin{aligned} \vec\nabla V&=\langle 4yz,4xz,4xy \rangle \\ \vec\nabla g&=\langle 4x,2y,1 \rangle \end{aligned} So the Lagrange equations, ∇⃗V=λ∇⃗g\vec\nabla V=\lambda\vec\nabla g, are: 4yz=λ4x(1)4xz=λ2y(2)4xy=λ(3)\begin{aligned} 4yz&=\lambda 4x \qquad \text{(1)} \\ 4xz&=\lambda 2y \qquad \text{(2)} \\ 4xy&=\lambda \qquad \text{(3)} \end{aligned} We multiply (1) by xx, (2) by yy, (3) by zz and equate the left hand sides: 4xyz=λ4x2=λ2y2=λz 4xyz=\lambda 4x^2=\lambda 2y^2=\lambda z This says x2=z4x^2=\dfrac{z}{4} and y2=z2y^2=\dfrac{z}{2}. We plug these into the constraint: 16=z+2x2+y2=z+z2+z2=2z 16=z+2x^2+y^2=z+\dfrac{z}{2}+\dfrac{z}{2}=2z So z=8x=2y=2 z=8 \qquad x=\sqrt{2} \qquad y=2 So the length, width, height and volume are: L=2x=22W=2y=4H=z=8V=LWH=642\begin{aligned} L=2x=2\sqrt{2} \qquad &W=2y=4 \qquad H=z=8 \\ V=&LWH=64\sqrt{2} \end{aligned}

    [×]

    Remark

    Notice that this is the same problem as previously solved. There we eliminated a variable. Here we used Lagrange mumtipliers.

    [×]
  17. Find the radius of the sphere with its center at the origin which is tangent to the plane 2x−2y−z=182x-2y-z=18.
    Solve by using a Lagrange multiplier.
    Rotate the second plot with your mouse.

    x_sph2plane
    1

    Hint

    The radius of the sphere is the shortest distance to the plane. It is easier to minimize the square of the distance.

    [×]

    Answer

    The radius is r=6r=6.

    [×]

    Solution

    The radius of the sphere is the shortest distance to the plane. We minimize the square of the distance: f=D2=x2+y2+z2 f=D^2=x^2+y^2+z^2 subject to the constraint that the point is on the plane: g=2x−2y−z=18 g=2x-2y-z=18 The gradients are ∇⃗f=⟨2x,2y,2z⟩∇⃗g=⟨2,−2,−1⟩ \vec\nabla f=\langle 2x,2y,2z \rangle \qquad \vec\nabla g=\langle 2,-2,-1 \rangle The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 2x=2λ2y=−2λ2z=−λ 2x=2\lambda \qquad 2y=-2\lambda \qquad 2z=-\lambda Consequently, λ=x=−y=−2z\lambda=x=-y=-2z. Plug these into the constraint: 18=2x−2y−z=−4z−4z−z=−9z⟹z=−2 18=2x-2y-z=-4z-4z-z=-9z \qquad \Longrightarrow \qquad z=-2 So x=4x=4, y=−4y=-4 and the point of tangency is (x,y,z)=(4,−4,−2)(x,y,z)=(4,-4,-2). So the radius is: r=42+42+22=6 r=\sqrt{4^2+4^2+2^2}=6

    tj 

    [×]

    Remark

    Notice that this is effectively the same problem as previously solved because in both cases we are minimizing the distance from the origin to the plane. There we eliminated a variable. Here we used Lagrange mumtipliers.

    [×]
  18. Find the point(s) on the hyperbolic cylinder z2−y2=1z^2-y^2=1 closest to the origin.

    Hint

    Only find the real solutions when solving the Lagrange equations.

    [×]

    Answer

    (0,0,−1)(0, 0, -1) and (0,0,1)(0, 0, 1)

    [×]

    Solution

    We minimize the square of the distance function from the origin using f=D2=x2+y2+z2f=D^2 = x^2+y^2+z^2 on the hyperbolic cylinder g=z2−y2=1g= z^2-y^2=1. The gradients of the functions are ∇⃗f=⟨2x,2y,2z⟩∇⃗g=⟨0,−2y,2z⟩\begin{aligned} \vec\nabla f &=\langle 2x, 2y, 2z\rangle \\ \vec\nabla g&=\langle0,-2y,2z\rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 2x=0(1)2y=−2λy(2)2z=2λz(3)\begin{aligned} 2x&=0 \qquad \text{(1)} \\ 2y&=-2\lambda y \qquad \text{(2)} \\ 2z&=2\lambda z \qquad \text{(3)} \end{aligned} Equation (1) yields x=0x = 0. Factoring equation (2) gives: y(2+2λ)=0 y \left(2+2\lambda \right)=0 The two solution are y=0y=0 and λ=−1\lambda=-1.
    Case 1: Plugging y=0y=0 into the constraint equation z2−y2=1z^2-y^2=1, gives: z=±1 z = \pm 1 and equation (3) gives λ=1\lambda=1..
    Case 2: Plugging λ=−1\lambda=-1 into equation (3), gives z=−zz=-z which says z=0z=0. Then the constraint z2−y2=1z^2-y^2=1 gives −y2=1-y^2=1 which only has imaginary solutions.
    So the two critical points are: (0,0,−1)and(0,0,1) (0, 0, -1) \quad\text{and} \quad (0, 0, 1) which are at a distance D=02+02+12=1D=\sqrt{0^2+0^2+1^2}=1 from the origin. Any other point would have z2=1+y2≥1z^2=1+y^2 \ge 1 and so a bigger distance.

    wll 

    [×]
  19. Find the largest and smallest values of the function f(x,y)=xyf(x,y)=xy on the ellipse x218+y28=1\dfrac{x^2}{18}+\dfrac{y^2}{8}=1.

    x_xy_on_ellipse

    Hint

    Find the gradients of both functions.

    [×]

    Answer

    The smallest value of f(x,y)=xyf(x,y) = xy along the ellipse is −6-6.
    The largest value of f(x,y)=xyf(x,y) = xy along the ellipse is 66.

    [×]

    Solution

    We maximize and minimize the function f=xyf=xy with the constraint of the ellipse g=x218+y28=1g= \dfrac{x^2}{18}+\dfrac{y^2}{8}=1.

    The gradients of the functions are ∇⃗f=⟨y,x⟩∇⃗g=⟨x9,y4⟩ \vec\nabla f=\langle y,x \rangle \qquad \vec\nabla g=\left\langle \dfrac{x}{9},\dfrac{y}{4} \right\rangle The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are y=λx9x=λy4 y=\dfrac{\lambda x}{9} \qquad x=\dfrac{\lambda y}{4} Solving for λ\lambda in each equation yields λ=9yx=4xy⟹x2=9y24(*) \lambda = \dfrac{9y}{x} = \dfrac{4x}{y} \implies x^2=\dfrac{9y^2}{4} \qquad \text{(*)} Plugging this result into the constraint equation gives: 1189y24+y28=1y28+y28=1y2=4y=±2\begin{aligned} \dfrac{1}{18}\dfrac{9y^2}{4} + \dfrac{y^2}{8}&=1 \\ \dfrac{y^2}{8} + \dfrac{y^2}{8}&=1 \\ y^2 &= 4 \\ y &= \pm 2 \end{aligned} Plugging these values of yy back into (*) gives x=±3x = \pm 3. This results in 4 critical points along the elliptical constraint: (3,2)(3,−2)(−3,2)(−3,−2) (3,2) \qquad (3,-2) \qquad (-3,2) \qquad (-3,-2) We compute the values of f=xyf=xy: f(3,2)=f(−3,−2)=6andf(3,−2)=f(−3,2)=−6 f(3,2)=f(-3,-2)=6 \quad \text{and} \quad f(3,-2)=f(-3,2)=-6 Consequently, the smallest value of ff is −6-6 and largest values is 66.

    wll 

    [×]
  20. Find the largest and smallest values of the function f(x,y,z)=xyzf(x,y,z)=xyz on the ellipsoid x218+y28+z22=1\dfrac{x^2}{18}+\dfrac{y^2}{8}+\dfrac{z^2}{2}=1.

    In the plot, the ellipsoid is red, the level sets of ff where f>0f \gt 0 are blue, and the level sets of ff where f<0f \lt 0 are yellow.

    x_xyz_on_ellipsoid
    1

    Hint

    Do the previous problem first. The solutions are similar and the proccess is identical.

    [×]

    Answer

    The smallest value is −463-\dfrac{4\sqrt{6}}{3}.
    The largest value is 463\dfrac{4\sqrt{6}}{3}.

    [×]

    Solution

    We maximize and minimize the function f=xyzf=xyz with the constraint ellipsoid g=x218+y28+z22=1g = \dfrac{x^2}{18}+\dfrac{y^2}{8}+\dfrac{z^2}{2}=1.

    The gradients of the functions are ∇⃗f=⟨yz,xz,xy⟩∇⃗g=⟨x9,y4,z⟩ \vec\nabla f=\langle yz,xz,xy \rangle \qquad \vec\nabla g=\left\langle \dfrac{x}{9},\dfrac{y}{4},z \right\rangle The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are yz=λx9xz=λy4xy=λz yz=\dfrac{\lambda x}{9} \qquad xz=\dfrac{\lambda y}{4} \qquad xy=\lambda z We multiply the first equation by xx, the second by yy and the third by zz and equate them: xyz=λx29=λy24=λz2 xyz =\dfrac{\lambda x^2}{9} = \dfrac{\lambda y^2}{4} =\lambda z^2 These tell us: x2=9z2y2=4z2(*) x^2=9z^2 \qquad y^2=4z^2 \qquad \text{(*)} Plugging these equations into the constraint equation gives: 9z218+4z28+z22=13z22=1z=±23\begin{aligned} \dfrac{9z^2}{18} + \dfrac{4z^2}{8} + \dfrac{z^2}{2} &=1 \\ \dfrac{3z^2}{2} &=1 \\ z &= \pm\,\dfrac{\sqrt{2}}{\sqrt{3}} \end{aligned} Plugging these values of zz into (*) gives: x2=923⟹x=±6y2=423⟹y=±223\begin{aligned} x^2=9\dfrac{2}{3} &\implies x = \pm\,\sqrt{6} \\ y^2=4\dfrac{2}{3} &\implies y = \pm\,\dfrac{2\sqrt{2}}{\sqrt{3}} \end{aligned} Plugging positive and negative variations of (x,y,z)=(±6,±223,±23)(x,y,z)=\left(\pm\,\sqrt{6},\pm\,\dfrac{2\sqrt{2}}{\sqrt{3}}, \pm\,\dfrac{\sqrt{2}}{\sqrt{3}}\right) into f=xyzf=xyz give f=±463f=\pm\,\dfrac{4\sqrt{6}}{3}. So the a minimum value of ff is −463-\dfrac{4\sqrt{6}}{3} and the maximum value is 463\dfrac{4\sqrt{6}}{3}.

    wll,tj 

    [×]
  21. PY: x3d plot not showing.
  22. A 425cm425\,\text{cm} wire is cut into 22 pieces of lengths aa and bb. The piece of length aa is bent into a square. The piece of length bb is bent into a rectangle whose length is twice its width. What are aa and bb which minimize or maximize the total area enclosed by both pieces? (Allow for the case that aa or bb is zero.)

    x_fold_sq_rect_2x1

    Hint

    Express the total area as a function of its segment lengths aa and bb. Find the critical points where aa and bb are both non-zero and where one is zero. Then make a table of values.

    [×]

    Answer

    Minimized Area =106252≈5312.5=\dfrac{10625}{2}\approx 5312.5
    at a=200cma=200\,\text{cm} and b=225cmb=225\,\text{cm}

    Maximized Area =18062516≈11289.=\dfrac{180625}{16}\approx11289.
    at a=425cma=425\,\text{cm} and b=0cmb=0\,\text{cm}

    [×]

    Solution

    We represent the length of the original wire as a sum of its parts: a+b=425(1) a + b = 425 \qquad \text{(1)} To minimize the total area of the shapes, we find the area of each shape as a function of the wire segment lengths. First the segment of length aa is folded into a square of side ss: Perimetera=a=4s⇒s=a4Areaa=s2=a216\begin{aligned} \text{Perimeter}_{a} &= a = 4s \quad \Rightarrow \quad s = \dfrac{a}{4}\\ \text{Area}_{a} &= s^2= \dfrac{a^2}{16} \end{aligned} And now the segment of length bb is folded into a rectangle of width ww and length 2w2w: Perimeterb=b=2w+w+2w+w=6w⇒w=b6Areab=2w⋅w=2w2=b218\begin{aligned} \text{Perimeter}_{b} &= b = 2w + w + 2w + w = 6w \quad \Rightarrow \quad w = \dfrac{b}{6}\\ \text{Area}_{b} &= 2w \cdot w = 2w^2= \dfrac{b^2}{18} \end{aligned} The total area is then the sum of two areas. A=a216+b218(∗) A = \dfrac{a^2}{16} + \dfrac{b^2}{18} \qquad (*) We apply the constraint (1) by plugging b=425−ab=425-a into our total area function: A=a216+(425−a)218 A= \dfrac{a^2}{16} + \dfrac{(425-a)^2}{18} We set the derivative equal to 00 and solve of aa: dAda=a8−425−a9=0⇒9a=8(425−a)⇒17a=3400⇒a=200\begin{aligned} &\dfrac{dA}{da} = \dfrac{a}{8} - \dfrac{425-a}{9}=0 \quad \Rightarrow \quad 9a=8(425-a) \\ &\Rightarrow \quad 17a=3400 \quad \Rightarrow \quad a= 200 \end{aligned} Plugging back this gives: b=425−a=425−200=225 b=425-a=425-200=225 We test the critical points and the endpoint cases (where aa or bb are zero) to see which is a minimum or maximum. We plug aa and bb into (*) and make a table of values:

    aa bb Area Classification
    200200 225225 106252≈5312.5\dfrac{10625}{2}\approx 5312.5 Local Minimum
    00 425425 18062518≈10035.\dfrac{180625}{18}\approx10035. Neither
    425425 00 18062516≈11289.\dfrac{180625}{16}\approx11289. Local Maximum

    wll,tj 

    [×]

    Remark

    Note: The maximized area occurs when the square has all of the wire. If you think of a puddle of water on a piece of glass, it will contract into a circle which has maximum area and minimum perimeter. Here we are constrained to a square and a rectangle and the square with no rectangle is the closest thing to a circle.

    [×]
  23. A 166cm166\,\text{cm} wire is cut into 33 pieces of lengths aa, bb and cc. The piece of length aa is bent into a square. The piece of length bb is bent into a rectangle whose length is twice its width. The piece of length cc is bent into a rectangle whose length is 33 times its width. What are aa, bb and cc which minimize and maximize the total area enclosed by all 33 pieces? (Allow for cases where aa, bb, or cc are zero.)

    x_fold_sq_rect_2x1_rect_3x1

    Hint

    Express the total area as a function of its segment lengths aa, bb and cc. Find the critical points where aa, bb and cc are all non-zero, where one is zero and where two are zero. Then make a table of values.

    [×]

    Answer

    Maximized Area:
    a=166a = 166 \quad b=0b = 0 \quad c=0c = 0
    Minimized Area:
    a=48a = 48 \quad b=54b = 54 \quad c=64c = 64

    [×]

    Solution

    We express the length of the wire as the sum of its parts: L(a,b,c)=a+b+c=166 L(a,b,c) = a + b +c = 166 To minimize the total area of the shapes, we find the area of each shape as a function of the wire segment lengths.
    The piece of length aa is folded into a square of side ss: Perimetera=a=4sAreaa=s2=a216\begin{aligned} \text{Perimeter}_{a} &= a = 4s \\ \text{Area}_{a} &= s^2= \dfrac{a^2}{16} \end{aligned} The piece of length bb is folded into a rectangle of width ww and length 2w2w: Perimeterb=b=2w+w+2w+w=6wAreab=2w⋅w=2w2=b218\begin{aligned} \text{Perimeter}_{b} &= b = 2w + w + 2w + w = 6w \\ \text{Area}_{b} &= 2w \cdot w = 2w^2= \dfrac{b^2}{18} \end{aligned} The piece of length 33 is folded into a rectangle of width WW and length 3W3W: Perimeterc=c=3W+W+3W+W=8WAreac=3W⋅W=3W2=3c264\begin{aligned} \text{Perimeter}_{c} &= c = 3W + W +3W + W = 8W \\ \text{Area}_{c} &= 3W \cdot W = 3W^2= \dfrac{3c^2}{64} \end{aligned} The total area is then found by adding the three areas: A=a216+b218+3c264\begin{aligned} A &= \dfrac{a^2}{16} + \dfrac{b^2}{18}+ \dfrac{3c^2}{64}\\ \end{aligned} The gradients of the functions are ∇⃗A=(a8,b9,3c32)∇⃗L=(1,1,1) \vec\nabla A=\left(\dfrac{a}{8}, \dfrac{b}{9}, \dfrac{3c}{32}\right) \qquad \vec\nabla L=\left(1,1,1\right) The Lagrange equations, ∇⃗A=λ∇⃗L\vec\nabla A=\lambda\vec\nabla L, are a8=λ(1)b9=λ(2)3c32=λ(3)\begin{aligned} \dfrac{a}{8}&=\lambda \qquad \text{(1)} \\ \dfrac{b}{9}&=\lambda \qquad \text{(2)} \\ \dfrac{3c}{32}&=\lambda \qquad \text{(3)} \end{aligned} Solving for bb and cc in terms of aa gives: a8=b9⟹b=9a8a8=3c32⟹c=4a3\begin{aligned} \dfrac{a}{8}&=\dfrac{b}{9} &\implies b=\dfrac{9a}{8} \\ \dfrac{a}{8}&=\dfrac{3c}{32} &\implies c=\dfrac{4a}{3} \end{aligned} Plugging these results into the constraint L=166L=166 gives: a+9a8+4a3=16683a24=166a=48\begin{aligned} a + \dfrac{9a}{8} + \dfrac{4a}{3} &= 166 \\ \dfrac{83a}{24} &= 166 \\ a &= 48 \end{aligned} And using this value in previous equations yields: b=54c=64 b = 54 \qquad c = 64

    So a critical point is (48,54,64)\left(48,54, 64\right).

    We also need to check the boundaries. The plane, a+b+c=166a+b+c=166, intersects the first quadrant in a triangle. So we check the 33 edges (where one of aa, bb or cc is zero and we need to extremize relative to the 22 other variables) and the 33 vertices (where two are zero and other one is 166166). For the edges, we compute:

    First, when a=0a=0:

    x_3piecewire

    A=b218+3c264L(b,c)=b+c=166∇⃗A=(b9,3c32)∇⃗L=(1,1)\begin{aligned} A &= \dfrac{b^2}{18}+ \dfrac{3c^2}{64} \qquad &L(b,c) &= b + c = 166\\ \vec\nabla A&=\left(\dfrac{b}{9}, \dfrac{3c}{32}\right) &\vec\nabla L&=(1,1) \end{aligned} The Lagrange equations and the constraint lead to: b=27c3227c32+c=166c=531259b=448259\begin{aligned} b &= \dfrac{27c}{32}\\ \dfrac{27c}{32} + c &= 166\\ c &= \dfrac{5312}{59} \qquad b=\dfrac{4482}{59} \end{aligned} Then when b=0b=0 f(a,b,c)=a+c=166A=a216+3c264∇⃗A=(a8,3c32)a=3c43c4+c=166c=6647a=4987\begin{aligned} f(a,b,c) &= a + c = 166\\ A &= \dfrac{a^2}{16} + \dfrac{3c^2}{64}\\ \vec\nabla A&=\left(\dfrac{a}{8}, \dfrac{3c}{32}\right)\\ a &= \dfrac{3c}{4}\\ \dfrac{3c}{4} + c &= 166\\ c &= \dfrac{664}{7} \qquad a=\dfrac{498}{7} \end{aligned} Then when c=0c=0 f(a,b,c)=a+b=166A=a216+b218∇⃗A=(a8,b9)a=8b98b9+b=66b=149417a=132817\begin{aligned} f(a,b,c) &= a + b = 166\\ A &= \dfrac{a^2}{16} + \dfrac{b^2}{18}\\ \vec\nabla A&=\left(\dfrac{a}{8}, \dfrac{b}{9}\right)\\ a &= \dfrac{8b}{9}\\ \dfrac{8b}{9} + b &= 66\\ b &= \dfrac{1494}{17} \qquad a=\dfrac{1328}{17} \end{aligned} We make a table of values:

    aa bb cc Area Classification
    4848 5454 6464 498498 Local Minimum
    00 448259\dfrac{4482}{59} 531259\dfrac{5312}{59} 4133459≈700.6\dfrac{41334}{59}\approx700.6 Neither
    4987\dfrac{498}{7} 00 6647\dfrac{664}{7} 2066728≈738.1\dfrac{20667}{28}\approx738.1 Neither
    132817\dfrac{1328}{17} 149417\dfrac{1494}{17} 00 1377817≈810.5\dfrac{13778}{17}\approx810.5 Neither
    00 00 166166 2066716≈1292.\dfrac{20667}{16}\approx1292. Neither
    00 166166 00 137789≈1531.\dfrac{13778}{9}\approx1531. Neither
    166166 00 00 68894≈1722.\dfrac{6889}{4}\approx1722. Local Maximum

    wll,tj 

    [×]

    Remark

    Note: The maximized area occurs when the square has all of the wire. If you think of a puddle of water on a piece of glass, it will contract into a circle which has maximum area and minimum perimeter. Here we are constrained to a square and a rectangle and the square with no rectangle is the closest thing to a circle.

    [×]
  24. A farmer wants to fence in a grazing field and split it into two pieces for cows and horses. The field will be a rectangle surrounded by fence with a divider fence parallel to one side. What are the length, width and area of the field which maximize the total area if the total length of fence is 30003000 meters?

    x_fence_1div

    Hint

    Write the area and perimeter in terms of xx and yy. Then set up the Lagrange equations.

    [×]

    Answer

    The dimensions that maximize the area are:
    x=750x = 750 and y=500y = 500
    where yy is the length as the partition. These dimensions produce a total area of A=375000A=375000.

    [×]

    Solution

    The total fence length is written as a function of width, xx, and the height yy including the partition. So the perimeter constraint is: P(x,y)=2x+3y=3000 P(x,y) = 2x+3y = 3000 The total enclosed area is: A(x,y)=xy A(x,y) = xy The gradients of the functions are ∇⃗A=⟨y,x⟩∇⃗P=⟨2,3⟩ \vec\nabla A=\langle y,x \rangle \qquad \vec\nabla P=\langle 2,3 \rangle The Lagrange equations, ∇⃗A=λ∇⃗P\vec\nabla A=\lambda\vec\nabla P, are y=λ2(1)x=λ3(2)\begin{aligned} y&=\lambda 2 \qquad \text{(1)} \\ x&=\lambda 3 \qquad \text{(2)} \end{aligned} Dividing equation (1) by equation (2) yields y=23xy=\dfrac{2}{3}x.
    Plugging this into the constrained equation gives: 2x+3(23x)=30002x+2x=3000x=750\begin{aligned} 2x+3\left(\dfrac{2}{3}x\right) &= 3000 \\ 2x + 2x &= 3000 \\ x &= 750 \end{aligned} We plug this the formula for yy to get y=500y = 500. These two dimensions produce a total area of A=750⋅500=375000A=750\cdot500=375000.

    This area is the maximum because yy is constrained to 0≤y≤10000 \le y \le 1000 and the area is 00 at both endpoints while the only interior critical point gives a positive area.

    wll,tj 

    [×]
  25. A box manufacturer wants to make a rectangular solid cardboard box with a vertical cardboard divider which splits the interior into 22 compartments, each containing 12000cm312\,000\,\text{cm}^3. Because of folding, the top and bottom will have 22 layers of cardboard. What are the dimensions of the box which uses the least amount of cardboard?

    x_box_div

    Hint

    Express the volume and surface area in terms of the box's length, width and height. Be sure to include the area of the partition. The volume is the constraint. Minimize the surface area.

    [×]

    Answer

    x=30cmy=20cmz=40cm x=30\,\text{cm} \quad y=20\,\text{cm} \quad z=40\,\text{cm}

    [×]

    Solution

    Let xx and yy be the length and width of the box and let zz be the height. There are 44 layers of cardboard on the top and bottom with area xyxy. There are 22 layers of cardboard in the front and back with area xzxz. Finally, there are 33 layers of cardboard on the left, the right and the partition with area yzyz. All together, the area is: A(x,y,z)=4xy+2xz+3yz A(x, y, z) = 4xy + 2xz + 3yz And the total enclosed volume is: V(x,y,z)=xyz=2⋅12000=24000 V(x,y,z) = xyz = 2\cdot12\,000 = 24\,000 The gradients of the functions are ∇⃗A=⟨4y+2z,4x+3z,2x+3y⟩∇⃗V=⟨yz,xz,xy⟩ \vec\nabla A=\langle 4y+2z,4x+3z,2x+3y \rangle \qquad \vec\nabla V=\langle yz,xz,xy \rangle The Lagrange equations, ∇⃗A=λ∇⃗V\vec\nabla A=\lambda\vec\nabla V, are 4y+2z=λyz(1)4x+3z=λxz(2)2x+3yx=λxy(3)\begin{aligned} 4y+2z&=\lambda yz \qquad \text{(1)} \\ 4x+3z&=\lambda xz \qquad \text{(2)} \\ 2x+3yx&=\lambda xy \qquad \text{(3)} \end{aligned} We multiply equation (1) by xx, equation (2) by yy, and equation (3) by zz. We then equate and solve: λxyz=2xz+4xy=3yz+4xy=3yz+2xz⟹2x=3y4y=2z\begin{aligned} \lambda xyz&=2xz+4xy = 3yz+4xy = 3yz+2xz \\ &\implies \quad 2x = 3y \qquad 4y = 2z \end{aligned} Substituting these values of xx and zz into the volume constraint gives: V=xyz=(3y2)y(2y)=3y3=24000\begin{aligned} V &= xyz = \left(\dfrac{3y}{2}\right) y (2y) \\ &= 3y^3=24\,000 \\ \end{aligned} So y=20cmy=20\,\text{cm}, and then x=30cmx=30\,\text{cm} and z=40cmz=40\,\text{cm}.

    wll 

    [×]
  26. Find the maximum value of f=xyf=xy on the line x+y=6x+y=6.

    x_maxProd_on_sum_prob

    Hint

    To see the critical point (x,y)=(a,b)(x,y)=(a,b) produces a maximum, look at the values of ff at the nearby point x=a+δx=a+\delta and y=b−δy=b-\delta.

    [×]

    Answer

    The maximum value of ff is f(3,3)=9f(3,3)=9.

    [×]

    Solution

    We maximize f=xyf=xy with the constraint set g=x+y=6g=x+y=6. The gradients of the functions are ∇⃗f=⟨y,x⟩and∇⃗g=⟨1,1⟩ \vec\nabla f=\langle y,x \rangle \quad \text{and} \quad \vec\nabla g=\langle 1,1 \rangle

    x_maxProd_on_sum_sol

    The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are y=λx=λ⟹x=y\begin{aligned} y = \lambda \qquad x = \lambda \quad \implies \quad x=y \end{aligned} Plugging this into the constraint equation gives: x+x=6x=3⟹y=3\begin{aligned} x + x &= 6 \\ x &= 3 \quad \implies \quad y=3 \end{aligned} So the critical point is (x,y)=(3,3)(x,y)=(3,3) where f(3,3)=9f(3,3)=9. To see this is a maximum, we look at a near by point with x=3+δx=3+\delta and y=3−δ y=3-\delta. Then: f(x,y)=(3+δ)(3−δ)=9−δ2<9 f(x,y)=(3+\delta)(3-\delta)=9-\delta^2 \lt 9 So the maximum point of ff on the line gg is f(3,3)=9f(3,3)=9.

    wll,tj 

    [×]
  27. Find the minimum value of f=x+yf=x+y on the branch of the hyperbola xy=9xy=9 in the first quadrant. Discuss the geometrical relation between this problem and the previous problem where the constraint and the extremal are interchanged.

    x_maxSum_on_prod_prob

    Hint

    To check the critical point is a minimum, look at the values of ff at the largest and smallest value of xx that are allowed.

    [×]

    Answer

    The minimum value of ff is f(3,3)=6f(3, 3)=6. The relationship is discussed in the remark.

    [×]

    Solution

    We minimizef=x+yf=x+y with the constraint set g=xy=9g=xy=9. The gradients of the functions are ∇⃗f=⟨1,1⟩and∇⃗g=⟨y,x⟩ \vec\nabla f=\langle 1,1 \rangle \quad \text{and} \quad \vec\nabla g=\langle y,x \rangle

    x_maxSum_on_prod_sol

    The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 1=λy1=λx⟹x=y\begin{aligned} 1 = \lambda y \qquad 1 = \lambda x \quad \implies \quad x=y \end{aligned} Plugging this into the constraint equation gives: x2=9x=±3⟹y=x=±3\begin{aligned} x^2 &= 9 \\ x &= \pm 3 \quad \implies \quad y=x=\pm3 \end{aligned} Since we want the point in the first quadrant, the critical points is (3,3)(3, 3). To see this is a minimum, we notice that when xx is very large, then y=9xy=\dfrac{9}{x} is close to 00 and f(x,y)f(x,y) is very large. Similarly, when xx is close to 00, then y=9xy=\dfrac{9}{x} is very large and f(x,y)f(x,y) is also very large. So there must be a minimum in between and (3,3)(3, 3) is the only critical point. So, the minimum point of ff on the hyperbola gg is f(3,3)=6f(3,3)=6.

    wll,tj 

    [×]

    Remark

    This and the previous problem are geometricaly dual in their solution. In both scenarios, the solution (minimum or maximum) is the point of tangency of a line and a hyperbola.

    For example, the constraint xy=9xy=9 of this problem is a specific hyperbola. The level sets of the extremal function f=x+yf=x+y are the family of all lines with slope m=−1m=-1. The solution of this problem is the point where one of these lines is tangent to the hyperbola xy=9xy=9.

    Similarly, in the previous problem, the constraint x+y=6x + y = 6 is a specific line. The level sets of the extremal function f=xyf=xy are the family of all hyperbolas of the form y=Cxy=\dfrac{C}{x}. The solution of that problem is the point where one of these hyperbolas is tangnet to the line x+y=6x+y=6.

    The differ because the constraint and the extremal are interchanged.

    [×]
  28. Find the maximum volume of an cylindrical aluminum can with lids whose surface area is 216πcm2216\pi\,\text{cm}^2.

    x_can_maxVol432_fixSurf216

    Hint

    Construct functions for the volume and surface area of the can. Remember the area of the two ends. Treat rr and hh as you would xx and yy.

    [×]

    Answer

    A radius r=6r=6 and height h=12h=12 produces a maximized volume of V=432πV=432\pi.

    [×]

    Solution

    The curved part of the can has area 2πrh2\pi rh while the top and bottom have area πr2\pi r^2. So we want the total area to be A=2πrh+2πr2=216πA=2\pi rh+2\pi r^2=216\pi. We maximize the volume function, V=πr2hV=\pi r^2 h subject to the area constraint. The gradients of the functions are ∇⃗V=⟨2πrh,πr2⟩∇⃗A=⟨2πh+4πr,2πr⟩\begin{aligned} \vec\nabla V&=\langle 2\pi rh,\pi r^2 \rangle \\ \vec\nabla A&=\langle 2\pi h + 4\pi r, 2\pi r \rangle \end{aligned} The Lagrange equations, ∇⃗V=λ∇⃗A\vec\nabla V=\lambda\vec\nabla A, are 2πrh=λ(2πh+4πr)(1)πr2=λ2πr(2)\begin{aligned} 2\pi rh&=\lambda (2\pi h + 4\pi r) \qquad &\text{(1)} \\ \pi r^2&=\lambda 2\pi r \qquad &\text{(2)} \end{aligned} Dividing equation (1) by equation (2) gives: 2hr=hr+22h=h+2rh=2r\begin{aligned} \dfrac{2h}{r}&=\dfrac{h}{r}+2 \\ 2h&=h+2r \\ h &= 2r \end{aligned} Substituting this into the area constraint gives 2πr2+2πr(2r)=216π6πr2=216πr=6\begin{aligned} 2 \pi r^2 + 2 \pi r(2r)&=216\pi \\ 6\pi r^2&=216\pi\\ r&=6 \end{aligned} Therefore the critical point r=6r=6 , h=12h=12 produces a maximized volume of V=432πV=432\pi.

    wll,tj 

    [×]
  29. Find the minimum surface area of a cylindrical aluminum can with lids whose volume is 432πcm3432\pi\,\text{cm}^3. Discuss the geometrical relation between this problem and the previous problem.

    x_can_minSurf216_fixVol432

    Hint

    Construct functions for the volume and surface area of the can. Remember the area of the two ends. Treat rr and hh as you would xx and yy.

    [×]

    Answer

    The radius r=6r=6 and height h=12h=12 produces a minimized surface area A=216πA=216\pi.

    [×]

    Solution

    We minimize the surface area function, A=2πrh+2πr2A= 2 \pi rh+2 \pi r^2 , with the constraint of a fixed volume, V=πr2h=432πV=\pi r^2 h=432\pi.
    The gradients of the functions are ∇⃗A=⟨2πh+4πr,2πr⟩∇⃗V=⟨2πrh,πr2⟩\begin{aligned} \vec\nabla A&=\langle 2\pi h+4\pi r, 2\pi r \rangle \\ \vec\nabla V&=\langle 2\pi rh,\pi r^2 \rangle \end{aligned} The Lagrange equations, ∇⃗A=λ∇⃗V\vec\nabla A=\lambda\vec\nabla V, are 2πh+4πr=λ(2πrh)(1)2πr=λ(πr2)(2)\begin{aligned} 2\pi h+4\pi r&=\lambda (2\pi rh) \qquad &\text{(1)} \\ 2\pi r&=\lambda(\pi r^2) \qquad &\text{(2)} \end{aligned} Dividing equation (1) by equation (2) gives: hr+2=2hr2r+h=2hh=2r\begin{aligned} \dfrac{h}{r}+2&= \dfrac{2h}{r} \\ 2r + h&=2h \\ h &= 2r \end{aligned} Substituting this into the volume constraint gives πr2(2r)=432π2πr3=432πr=6\begin{aligned} \pi r^2 (2r)&=432\pi \\ 2\pi r^3&=432\pi\\ r&=6 \end{aligned} Therefore the critical point r=6r=6, h=12h=12 produces a minimized surface area A=216πA=216\pi.

    wll,tj 

    [×]

    Remark

    This and the previous problem are geometricaly dual in their solution. In both scenarios, the solution (minimum or maximum) is the point of tangency of two curves.

    In this problem, the constraint is the specific curve πr2h=432π\pi r^2 h=432\pi. The level sets of the extremal function A=2πrh+2πr2A=2 \pi rh+2 \pi r^2 are a family of curves. The solution of this problem is the point where one of the extremal family of curves is tangent to the constraint curve.

    Similarly, in the previous problem, the constraint is the specific curve 2πrh+2πr2=2162 \pi rh+2 \pi r^2=216. The level sets of the extremal function V=πr2hV=\pi r^2 h are a family of curves. The solution of that problem is the point where one of the extremal family of curves is tangnet to the constraint curve.

    The differ because the constraint and the extremal are interchanged.

    [×]
  30. Find the point(s) on the curve x2y=2x^2y=2 closest to the origin and give the distance from the origin.

    x_min_x^2y to 0

    Hint

    Minimize the square of the distance instead of the distance.

    [×]

    Answer

    The two points closest to the origin are: (2,1)(−2,1) (\sqrt{2}, 1) \qquad \quad (-\sqrt{2}, 1) which are at a distance of D=3D=\sqrt{3} from the origin.

    [×]

    Solution

    We minimize the square of the distance function from the origin using f=D2=x2+y2f=D^2 = x^2+y^2 on the constraint curve g=x2y=2g= x^2y=2. The gradients of the functions are ∇⃗f=⟨2x,2y⟩∇⃗g=⟨2xy,x2⟩\begin{aligned} \vec\nabla f&=\langle 2x, 2y \rangle \\ \vec\nabla g&=\langle 2xy,x^2 \rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 2x=2λxy(1)2y=λx2(2)\begin{aligned} 2x&=2\lambda xy \qquad &\text{(1)} \\ 2y&=\lambda x^2 \qquad &\text{(2)} \end{aligned} We divide (1) by (2) giving xy=2yx\dfrac{x}{y}=\dfrac{2y}{x} or x2=2y2x^2=2y^2. Plugging this into the constraint x2y=2x^2y=2 yields: 2y3=2y=1x=±2\begin{aligned} 2y^3&=2 \\ y &= 1 \\ x&=\pm\sqrt{2} \end{aligned} So the two minima points are: (2,1)(−2,1) (\sqrt{2}, 1) \qquad \quad (-\sqrt{2}, 1) which are at a distance D=22+12=3D=\sqrt{\sqrt{2}^2+1^2}=\sqrt{3} from the origin.

    wll,tj 

    [×]
  31. Find the point on the surface x4y2z=32x^4y^2z=32 closest to the origin.

    x_min_x^4y^2z_to_0

    Hint

    Minimize the square of the distance instead of the distance.

    [×]

    Answer

    The four points closest to the origin are: (Âą2,Âą2,1) (\pm2, \pm\sqrt{2}, 1) which are at a distance of D=7D=\sqrt{7} from the origin.

    [×]

    Solution

    We minimize the square of the distance function from the origin using f=D2=x2+y2+z2f=D^2 = x^2+y^2+z^2 on the constraint curve g=x4y2z=32g= x^4y^2z=32. The gradients of the functions are ∇⃗f=⟨2x,2y,2z⟩∇⃗g=⟨4x3y2z,2x4yz,x4y2⟩\begin{aligned} \vec\nabla f&=\langle 2x, 2y, 2z \rangle \\ \vec\nabla g&=\langle 4x^3y^2z,2x^4yz,x^4y^2 \rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 2x=4λx3y2z(1)2y=2λx4yz(2)2z=λx4y2(3)\begin{aligned} 2x&=4\lambda x^3y^2z \qquad &\text{(1)} \\ 2y&=2\lambda x^4yz \qquad &\text{(2)}\\ 2z&=\lambda x^4y^2 \qquad &\text{(3)} \end{aligned} We divide (1) by (3) giving xz=4zx\dfrac{x}{z}=\dfrac{4z}{x} or x2=4z2x^2=4z^2. We also divide (2) by (3) giving yz=2zy\dfrac{y}{z}=\dfrac{2z}{y} or y2=2z2y^2=2z^2. Plugging this into the constraint x4y2z=32x^4y^2z=32 yields: (16z4)(2z2)(z)=3232z7=32z=1x=±2y=±2\begin{aligned} (16z^4)(2z^2)(z)&=32 \\ 32z^7&=32 \\ z &=1 \\ x &=\pm2 \\ y &= \pm \sqrt{2} \end{aligned} So the the 44 minima points are: (±2,±2,1) (\pm2, \pm\sqrt{2}, 1) which are at a distance D=22+22+12=7D=\sqrt{2^2+\sqrt{2}^2+1^2}=\sqrt{7} from the origin.

    wll,tj 

    [×]
  32. Find the volume of the largest rectangular solid with 33 faces in the coordinate planes and the opposite vertex on the plane xa+yb+zc=1\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1 where a≥0a \ge 0, b≥0b \ge 0 and c≥0c \ge 0.

    x_rect_under_plane

    Hint

    Write the volume of the solid described in the question as a function of xx, yy, and zz Then you can use Lagrange equations to solve for the maximized volume.

    [×]

    Answer

    The maximized volume has a value of: V=abc27V=\dfrac{abc}{27}

    [×]

    Solution

    We maximize the volume, V=xyzV=xyz where (x,y,z)(x,y,z) is the vertex on the plane g=xa+yb+zc=1g=\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1. The gradients of the functions are ∇⃗V=⟨yz,xz,xy⟩∇⃗g=⟨1a,1b,1c⟩\begin{aligned} \vec\nabla V&=\left\langle yz, xz, xy\right\rangle \\ \vec\nabla g&=\left\langle\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c}\right\rangle \end{aligned} The Lagrange equations, ∇⃗V=λ∇⃗g\vec\nabla V=\lambda\vec\nabla g, are yz=λa(1)xz=λb(2)xy=λc(3)\begin{aligned} yz&=\dfrac{\lambda}{a} \qquad &\text{(1)} \\ xz&=\dfrac{\lambda}{b} \qquad &\text{(2)}\\ xy&=\dfrac{\lambda}{c} \qquad &\text{(3)} \end{aligned} We equate lambda in each equation, which yields the following equations: ayz=bxz=cxy⟹x=aybandz=cyb\begin{aligned} ayz=bxz=cxy \implies x=\dfrac{ay}{b} \quad \text{and} \quad z=\dfrac{cy}{b} \end{aligned} Substituting these values into the equation of the plane produces: ayba+yb+cybc=1yb+yb+yb=1y=b3\begin{aligned} \dfrac{\;\dfrac{ay}{b}\;}{a}+\dfrac{y}{b}+\dfrac{\;\dfrac{cy}{b}\;}{c}&=1 \\ \dfrac{y}{b}+\dfrac{y}{b}+\dfrac{y}{b}&=1 \\ y &= \dfrac{b}{3} \end{aligned} We use this value to solve for xx and zz. x=a(b3)b⟹x=a3z=c(b3)b⟹z=c3\begin{aligned} x=\dfrac{a\left(\dfrac{b}{3}\right)}{b} \implies x=\dfrac{a}{3} \\ z=\dfrac{c\left(\dfrac{b}{3}\right)}{b} \implies z=\dfrac{c}{3} \end{aligned} So the critical point is: (a3,b3,c3)\begin{aligned} \left(\dfrac{a}{3},\dfrac{b}{3},\dfrac{c}{3}\right) \end{aligned} With a corresponding maximized volume of V=a3⋅b3⋅c3=abc27 V=\dfrac{a}{3} \cdot \dfrac{b}{3} \cdot \dfrac{c}{3} =\dfrac{abc}{27}

    wll,tj 

    [×]
  33. We would like to find the line y=f(x)=mx+by=f(x)=mx+b which best fits a collection of 2020 data points (x1,y1),(x2,y2),⋯,(x20,y20) (x_1,y_1), \quad (x_2,y_2), \quad \cdots, \quad (x_{20},y_{20}) using a quadratic fit. For each point, (xi,yi)(x_i,y_i), the vertical distance to the line is ∣yi−f(xi)∣|y_i-f(x_i)|. Click to toggle the vertical line segments whose lengths are ∣yi−f(xi)∣|y_i-f(x_i)|. To say we want a quadratic fit, means we want to find mm and bb so that the sum of the squared vertical distances S=∑i=120(yi−f(xi))2 S=\sum_{i=1}^{20} \left(y_i-f(x_i)\right)^2 is a minimum. Find formulas for mm and bb in terms of the xix_i's and yiy_i's. What happens if we have more than 2020 points?

    x_bestfit_pts

    Hint

    Note: The xix_i's and yiy_i's are constants. The variables are mm and bb. It may help to start with fewer points, say 33, so you can write out all the terms in the summation.

    [×]

    Answer

    For 20 samples, the formula for mm and bb are: m=20∑i=120xiyi−∑i=120xi∑i=120yi20∑i=120xi2−(∑i=120xi)2b=∑i=120yi∑i=120xi2−∑i=120xi∑i=120xiyi20∑i=120xi2−(∑i=120xi)2\begin{aligned} m&=\dfrac{\displaystyle 20\sum_{i=1}^{20}x_i y_i-\sum_{i=1}^{20}x_i\sum_{i=1}^{20}y_i} {\displaystyle 20\sum_{i=1}^{20} x_i^2 - \left(\sum_{i=1}^{20} x_i\right)^2} \\ b&=\dfrac{\displaystyle \sum_{i=1}^{20}y_i\sum_{i=1}^{20}x_i^2-\sum_{i=1}^{20}x_i\sum_{i=1}^{20}x_i y_i} {\displaystyle 20\sum_{i=1}^{20} x_i^2 - \left(\sum_{i=1}^{20} x_i\right)^2} \end{aligned} General Formula: m=n∑i=1nxiyi−∑i=1nxi∑i=1nyin∑i=1nxi2−(∑i=1nxi)2b=∑i=1nyi∑i=1nxi2−∑i=1nxi∑i=1nxiyin∑i=1nxi2−(∑i=1nxi)2\begin{aligned} m&=\dfrac{\displaystyle n\sum_{i=1}^{n}x_i y_i-\sum_{i=1}^{n}x_i\sum_{i=1}^{n}y_i} {\displaystyle n\sum_{i=1}^{n} x_i^2 - \left(\sum_{i=1}^{n} x_i\right)^2} \\ b&=\dfrac{\displaystyle \sum_{i=1}^{n}y_i\sum_{i=1}^{n}x_i^2-\sum_{i=1}^{n}x_i\sum_{i=1}^{n}x_i y_i} {\displaystyle n\sum_{i=1}^{n} x_i^2 - \left(\sum_{i=1}^{n} x_i\right)^2} \end{aligned}

    [×]

    Solution

    We treat SS as a function, S(m,b)S(m,b), where xix_{i} and yiy_{i} are constants. We substitute f(xi)f(x_i): S=∑i=120(yi−(mxi+b))2=∑i=120(yi−mxi−b)2\begin{aligned} S&=\sum_{i=1}^{20} \left(y_i-(mx_i+b)\right)^2 \\ &=\sum_{i=1}^{20} \left(y_i-mx_i-b\right)^2 \end{aligned} We take the partial derivatives with respect to mm: Sm=∑i=1202(yi−mxi−b)(−xi)=2m∑i=120xi2+2b∑i=120xi−2∑i=120xiyi\begin{aligned} S_m&=\sum_{i=1}^{20} 2\left(y_i-mx_i-b\right)(-x_i) \\ &= 2m\sum_{i=1}^{20} x_i^2+ 2b\sum_{i=1}^{20}x_i -2\sum_{i=1}^{20}x_i y_i \end{aligned} and with respect to bb: Sb=∑i=1202(yi−mxi−b)(−1)=2m∑i=120xi+40b−2∑i=120yi\begin{aligned} S_b&=\sum_{i=1}^{20} 2\left(y_i-mx_i-b\right)(-1) \\ &=2m\sum_{i=1}^{20} x_i + 40b -2\sum_{i=1}^{20}y_i \end{aligned} To simplify the computation, we introduce the notation: Σx=∑i=120xiΣy=∑i=120yiΣxx=∑i=120xi2Σxy=∑i=120xiyi \Sigma_x=\sum_{i=1}^{20} x_i \quad \Sigma_y=\sum_{i=1}^{20} y_i \quad \Sigma_{xx}=\sum_{i=1}^{20} x_i^2 \quad \Sigma_{xy}=\sum_{i=1}^{20}x_i y_i To minimize S(m,b)S(m,b) we set (half) the partial derivatives equal to 00: 12Sm=mΣxx+bΣx−Σxy=0(1)12Sb=mΣx+20b−Σy=0(2)\begin{aligned} \dfrac{1}{2}S_m &= m \Sigma_{xx} + b\Sigma_x -\Sigma_{xy}=0 \qquad &\text{(1)} \\ \dfrac{1}{2}S_b &= m\Sigma_x + 20b -\Sigma_y=0 \qquad &\text{(2)} \end{aligned} And we solve this system of linear equations for mm and bb. Solving equation (2) for bb gives: b=120Σy−m20Σx\begin{aligned} b &=\dfrac{1}{20}\Sigma_y - \dfrac{m}{20}\Sigma_x \end{aligned} Plugging this value of bb into equation (1) gives: mΣxx+(120Σy−m20Σx)Σx−Σxy=020mΣxx+ΣxΣy−mΣx2−20Σxy=0\begin{aligned} m\Sigma_{xx}+ \left(\dfrac{1}{20}\Sigma_y - \dfrac{m}{20}\Sigma_x\right)\Sigma_x -\Sigma_{xy}&=0 \\ 20m\Sigma_{xx}+\Sigma_x\Sigma_y -m{\Sigma_x}^2 -20\Sigma_{xy}&=0 \\ \end{aligned} We solve for mm: m(20Σxx−Σx2)=20Σxy−ΣxΣym=20Σxy−ΣxΣy20Σxx−Σx2\begin{aligned} m\left(20\Sigma_{xx}-{\Sigma_x}^2\right) &=20\Sigma_{xy}-\Sigma_x\Sigma_y\\ m&=\dfrac{20\Sigma_{xy}-\Sigma_x\Sigma_y}{20\Sigma_{xx}-{\Sigma_x}^2} \end{aligned} And plugging this value of mm back into the solution for bb gives: b=Σy20−Σx20(20Σxy−ΣxΣy20Σxx−Σx2)=20ΣyΣxx−ΣyΣx2−20ΣxΣxy+Σx2Σy20(20Σxx−Σx2)=20ΣyΣxx−20ΣxΣxy20(20Σxx−Σx2)=ΣyΣxx−ΣxΣxy20Σxx−Σx2\begin{aligned} b &=\dfrac{\Sigma_y}{20}-\dfrac{\Sigma_x}{20} \left(\dfrac{20\Sigma_{xy}-\Sigma_x\Sigma_y} {20\Sigma_{xx}-{\Sigma_x}^2}\right)\\ &=\dfrac{20\Sigma_y\Sigma_{xx}-\Sigma_y{\Sigma_x}^2-20\Sigma_x\Sigma_{xy}+{\Sigma_x}^2\Sigma_y} {20(20\Sigma_{xx}-{\Sigma_x}^2)} \\ &=\dfrac{20\Sigma_y\Sigma_{xx}-20\Sigma_x\Sigma_{xy}} {20(20\Sigma_{xx}-{\Sigma_x}^2)} \\ &=\dfrac{\Sigma_y\Sigma_{xx}-\Sigma_x\Sigma_{xy}} {20\Sigma_{xx}-{\Sigma_x}^2} \end{aligned} This produces the final solutions: m=20Σxy−ΣxΣy20Σxx−Σx2b=ΣyΣxx−ΣxΣxy20Σxx−Σx2 m=\dfrac{20\Sigma_{xy}-\Sigma_x\Sigma_y}{20\Sigma_{xx}-{\Sigma_x}^2} \qquad\qquad b=\dfrac{\Sigma_y\Sigma_{xx}-\Sigma_x\Sigma_{xy}}{20\Sigma_{xx}-{\Sigma_x}^2}


    General formulas are found by repeating the process with nn rather than 2020. m=nΣxy−ΣxΣynΣxx−Σx2b=ΣyΣxx−ΣxΣxynΣxx−Σx2 m=\dfrac{n\Sigma_{xy}-\Sigma_x\Sigma_y}{n\Sigma_{xx}-{\Sigma_x}^2} \qquad\qquad b=\dfrac{\Sigma_y\Sigma_{xx}-\Sigma_x\Sigma_{xy}}{n\Sigma_{xx}-{\Sigma_x}^2} where Σx=∑i=1nxiΣy=∑i=1nyiΣxx=∑i=1nxi2Σxy=∑i=1nxiyi \Sigma_x=\sum_{i=1}^n x_i \quad \Sigma_y=\sum_{i=1}^n y_i \quad \Sigma_{xx}=\sum_{i=1}^n x_i^2 \quad \Sigma_{xy}=\sum_{i=1}^n x_i y_i

    wll,tj 

    [×]
  34. Find the locations and values of the global maximum and minimum of the function f(x,y)=3x+4yf(x,y)=3x+4y on the circle x2+y2=1x^2+y^2=1.

    In the plot, the circle is red and the level sets of ff are blue.

    x_lines_tan_to_circ_prob
    1. Solve using a Lagrange multiplier.

      Hint

      Find all the critical points on the circle, then make a table of values to determine the minimum and maximum locations.

      [×]

      Answer

      Global Maximum: f=5at(x,y)=(35,45) f=5 \quad \text{at} \quad (x,y)=\left(\dfrac{3}{5},\dfrac{4}{5}\right) Global Minimum: f=−5at(x,y)=(−35,−45) f=-5 \quad \text{at} \quad (x,y)=\left(-\,\dfrac{3}{5},-\,\dfrac{4}{5}\right)

      [×]

      Solution

      We check for critical points on the boundary circle using the Lagrange method. The gradients of the functions are ∇⃗f=⟨3,4⟩∇⃗g=⟨2x,2y⟩\begin{aligned} \vec\nabla f &=\langle 3,4 \rangle \\ \vec\nabla g &=\langle 2x,2y \rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 3=2λx(1)4=2λy(2)\begin{aligned} 3 = 2\lambda x \qquad \text{(1)}\\ 4 = 2\lambda y \qquad \text{(2)} \end{aligned} We divide equation (1) by equation (2) to get 34=xy\dfrac{3}{4}=\dfrac{x}{y} which implies x=34yx=\dfrac{3}{4}y. We plug this into the contraint g=x2+y2=1g=x^2+y^2=1 (3y4)2+y2=1y2=1625y=±45\begin{aligned} \left(\dfrac{3y}{4}\right)^2+y^2&=1 \\ y^2&=\dfrac{16}{25} \\ y&=\pm \dfrac{4}{5} \end{aligned} We use this value to solve for xx: (Note they must have the same sign.) x=3(±45)4⟹x=±35\begin{aligned} x=\dfrac{3\left(\pm \dfrac{4}{5}\right)}{4} \implies x=\pm \dfrac{3}{5} \end{aligned} So our critical points are: (35,45)and(−35,−45)\begin{aligned} \left(\dfrac{3}{5}, \dfrac{4}{5}\right) \qquad \text{and} \qquad \left(-\,\dfrac{3}{5}, -\,\dfrac{4}{5}\right) \end{aligned} We make a table of values:

      xx yy ff
      35\dfrac{3}{5} 45\dfrac{4}{5} 55
      −35-\dfrac{3}{5} −45-\dfrac{4}{5} −5-5

      So the global maximum is f=5f=5 at (x,y)=(35,45)(x,y)=\left(\dfrac{3}{5},\dfrac{4}{5}\right) and the global minimum is f=−5f=-5 at (x,y)=(−35,−45)(x,y)=\left(-\,\dfrac{3}{5},-\,\dfrac{4}{5}\right).

      wll, tj 

      [×]

      Remark

      The Lagrange multiplier method is designed to find the point where a level set of the optimal function is tangent to the constraint set. Here the constraint is the circle (in red) and the level sets of the optimal function are the straignt lines, (in blue) which are tangent precisely at
      the maximum (x,y)=(.6,.8)(x,y)=(.6,.8) and
      the minimum (x,y)=(−.6,−.8)(x,y)=(-.6,-.8).

      x_lines_tan_to_circ_sol
      [×]
    2. Solve by parametrizing the constraint.

      Hint

      Parametrize the circle as (x,y)=(cos⁥θ,sin⁥θ)(x,y)=(\cos\theta,\sin\theta).

      [×]

      Answer

      Global Maximum: f=5at(x,y)=(35,45) f=5 \quad \text{at} \quad (x,y)=\left(\dfrac{3}{5},\dfrac{4}{5}\right) Global Minimum: f=−5at(x,y)=(−35,−45) f=-5 \quad \text{at} \quad (x,y)=\left(-\,\dfrac{3}{5},-\,\dfrac{4}{5}\right)

      [×]

      Solution

      We parametrize the circle as (x,y)=(cos⁡θ,sin⁡θ)(x,y)=(\cos\theta,\sin\theta). Then the extremal function becomes: f=3x+4y=3cos⁡θ+4sin⁡θ\begin{aligned} f&=3x+4y \\ &=3\cos\theta+4\sin\theta \\ \end{aligned} We set the derivative equal to 00 and solve for θ\theta: f′(θ)=−3sin⁡θ+4cos⁡θ=0tan⁡θ=4/3\begin{aligned} f'(\theta)=-3\sin\theta&+4\cos\theta=0 \\ \tan\theta&=4/3 \end{aligned} Since the tangent is positive, the angle must be in the 1st1^\text{st} or 3rd3^\text{rd} quadrants. So the solutions are: cos⁡θ1=35sin⁡θ1=45(x1,y1)=(35,45)cos⁡θ2=−35sin⁡θ2=−45(x2,y2)=(−35,−45)\begin{aligned} \cos\theta_1&=\dfrac{3}{5} &\sin\theta_1&=\dfrac{4}{5} &(x_1,y_1)&=\left(\dfrac{3}{5},\dfrac{4}{5}\right) \\ \cos\theta_2&=-\,\dfrac{3}{5} &\sin\theta_2&=-\,\dfrac{4}{5} &(x_2,y_2)&=\left(-\,\dfrac{3}{5},-\,\dfrac{4}{5}\right) \end{aligned} Finally, we evaluate the extremal at the points: f(x1,y1)=3⋅35+4⋅45=95+165=255=5f(x2,y2)=3⋅−35+4⋅−45=−95−165=−255=−5\begin{aligned} f(x_1,y_1) &=3\cdot\dfrac{3}{5}+4\cdot\dfrac{4}{5} \\ &=\dfrac{9}{5}+\dfrac{16}{5} =\dfrac{25}{5}=5 \\ f(x_2,y_2) &=3\cdot\dfrac{-3}{5}+4\cdot\dfrac{-4}{5} \\ &=-\dfrac{9}{5}-\dfrac{16}{5} =-\dfrac{25}{5}=-5 \end{aligned} So the global maximum is f(35,45)=5f\left(\dfrac{3}{5},\dfrac{4}{5}\right)=5 and the global minimum is f(−35,−45)=−5f\left(-\,\dfrac{3}{5},-\,\dfrac{4}{5}\right)=-5.

      tj 

      [×]
  35. An aquarium is to have a volume of 1620cm31620\,\text{cm}^3. Its base will be made of marble which costs $.50.50 per cm2\text{cm}^2. The front will be glass which costs $.10.10 per cm2\text{cm}^2. The sides and back will be aluminum which costs $.05.05 per cm2\text{cm}^2. What are the dimensions of the cheapest such aquarium?

    x_aquarium

    Hint

    Construct two functions, the cost of the aquarium (optimizing), and the volume of the aquarium (constraint) as functions of the dimensions.

    [×]

    Answer

    The cheapest dimensions of the aquarium are (length) x=6cmx = 6\,\text{cm}, (depth) y=9cmy = 9\,\text{cm}, and (height) z=30cmz=30\,\text{cm}

    [×]

    Solution

    We define xx,yy, and zz such that the area of the base is xyxy, the front is xzxz, and the sides are yzyz. We then create a function for the cost of the aquarium: C=(.5)xy+(.1)xz+2(.05)yz+(.05)xzC=.5xy+.15xz+.1yz\begin{aligned} C &= (.5)xy + (.1)xz + 2(.05)yz + (.05)xz \\ C &= .5xy + .15xz + .1yz \end{aligned} With the defined contraint volume of V=xyz=1620V = xyz = 1620, we minimize CC with contraint VV using the Lagrange method. The gradients of the functions are ∇⃗C=⟨.5y+.15z,.5x+.1z,.15x+.1y⟩∇⃗V=⟨yz,xz,xy⟩\begin{aligned} \vec\nabla C &=\langle .5y + .15z,.5x + .1z, .15x + .1y \rangle \\ \vec\nabla V &=\langle yz,xz,xy \rangle \end{aligned} The Lagrange equations, ∇⃗C=λ∇⃗V\vec\nabla C=\lambda\vec\nabla V, are .5y+.15z=λyz(1).5x+.1z=λxz(2).15x+.1y=λxy(3)\begin{aligned} .5y + .15z = \lambda yz \qquad \text{(1)}\\ .5x + .1z = \lambda xz \qquad \text{(2)}\\ .15x + .1y = \lambda xy \qquad \text{(3)} \end{aligned} We multiply equation (1) by xx, equation (2) by yy, and equation (3) by zz. We can then equate the left side of each equation: λxyz=.5xy+.15xz=.5xy+.1yz=.15xz+.1yz \lambda xyz=.5xy + .15xz = .5xy + .1yz = .15xz + .1yz We solve the first and second equations: .5xy+.15xz=.5xy+.1yz⟹x=23y.5xy+.1yz=.15xz+.1yz⟹z=103y\begin{aligned} .5xy + .15xz = .5xy + .1yz &\implies x = \dfrac{2}{3}y \\ .5xy + .1yz = .15xz + .1yz &\implies z= \dfrac{10}{3}y \end{aligned} Plugging these solutions for xx and zz into VV gives: (23y)(y)(103y)=1620y3=729y=9\begin{aligned} \left(\dfrac{2}{3}y\right)(y)\left(\dfrac{10}{3}y\right) &= 1620\\ y^3&=729 \\ y&=9 \end{aligned} Using this solution of yy yields x=6x=6 and z=30z=30. Therefore the cheapest dimensions of the aquarium are (6,9,30)(6,9,30)

    wll,tj 

    [×]
  36. Find the locations and values of the global maximum and minimum of the function f(x,y)=2x2−8x+y2−6yf(x,y)=2x^2-8x+y^2-6y inside or on the triangle with vertices (0,0)(0,0), (0,4)(0,4) and (4,4)(4,4).

    x_parab_on_tri

    Hint

    Find the critical points in the interior, the edges, and the verticies. Then compare the values at those points to determine the minimum and maximum of the function.

    [×]

    Answer

    The minimum of the function in the traingular region is f=−17f=-17 at (2,3)(2,3), and the maximum is f=0f=0 at (0,0)(0,0)

    [×]

    Solution

    To find the critical points inside the triangle, we set the partial derivatives of f=2x2−8x+y2−6yf=2x^2-8x+y^2-6y equal to 00 and solve: fx=4x−8=0⟹x=2fy=2y−6=0⟹y=3\begin{aligned} f_x = 4x - 8 = 0 \qquad \implies x = 2\\ f_y = 2y-6 = 0 \qquad \implies y=3 \end{aligned} From the plot, we see the critical point (2,3)(2,3) lies within the traingle.

    x_parab_on_tri_crit

    Next, we find all the critical points along the edges of the triangle region, represented by x=0x=0, y=4y=4, and y=xy=x. We set the partial derivatives equal to zero with each respective constraint. For x=0x=0: f=y2−6yfy=2y−6=0⟹y=3\begin{aligned} f &= y^2 - 6y \\ f_y &= 2y-6 &= 0 \qquad \implies y=3 \end{aligned} This implies that (0,3)(0,3) is a critical point on x=0x=0. Next, for y=4y=4: f=2x2−8x−8fx=4x−8=0⟹x=2\begin{aligned} f &= 2x^2-8x-8\\ f_x &= 4x - 8 &= 0 \qquad \implies x=2 \end{aligned} This implies that (2,4)(2,4) is a critical point on y=4y=4. Next, for y=xy=x: f=2x2−8x+x2−6x=3x2−14xfx=6x−14=0⟹x=73\begin{aligned} f &= 2x^2-8x + x^2 - 6x = 3x^2-14x \\ f_x &= 6x-14 &= 0 \qquad \implies x=\dfrac{7}{3} \end{aligned} This implies that (73,73)\left(\dfrac{7}{3}, \dfrac{7}{3}\right) is a critical point on y=xy=x.

    We make a table of values of ff for each critical point inside, on the edges and at each vertex.

    xx yy  ff
    22 33 −17-17
    00 33 −9-9
    22 44 −16-16
    73\dfrac{7}{3} 73\dfrac{7}{3} −493≈−16.3-\,\dfrac49{3}\approx -16.3
    00 00  00
    00 44 −8-8
    44 44 −8-8

    So from the table we can conclude that the minimum of the function in the traingular region is f=−17f=-17 at (2,3)(2,3), and the maximum is f=0f=0 at (0,0)(0,0)

    wll 

    [×]
  37. Find the locations and values of the global maximum and minimum of the function f(x,y)=x2+2y2−6x−8yf(x,y)=x^2+2y^2-6x-8y inside or on the rectangle with vertices (0,0)(0,0), (4,0)(4,0), (4,3)(4,3) and (0,3)(0,3).

    Hint

    Find the critical points in the interior, the edges, and the vertices. Then compare the values at those points to determine the minimum and maximum of the function.

    [×]

    Answer

    The minimum is f=−17f=-17 at (3,2)(3,2).
    The maximum is f=0f=0 at (0,0)(0,0).

    [×]

    Solution

    To find the critical points inside the rectangle, we set the partial derivatives of f=x2+2y2−6x−8yf=x^2+2y^2-6x-8y equal to 00 and solve: fx=2x−6=0⟹x=3fy=4y−8=0⟹y=2\begin{aligned} f_x = 2x - 6 = 0 \qquad \implies x = 3\\ f_y = 4y-8 = 0 \qquad \implies y=2 \end{aligned} The critical point (3,2)(3,2) lies within the rectangle 0≤x≤40 \le x \le 4 and 0≤y≤30 \le y \le 3.

    Next, we find all the critical points along the edges of the rectangle, which are x=0x=0, x=4x=4, y=0y=0, and y=3y=3. We restrict the function to each edge and then set the derivative equal to zero and solve.
    For x=0x=0: f=2y2−8yfy=4y−8=0⟹(x,y)=(0,2)\begin{aligned} f &= 2y^2-8y\\ f_y &= 4y-8 &= 0 \qquad \implies (x,y)=(0,2) \end{aligned} For y=0y=0: f=x2−6xfx=2x−6=0⟹(x,y)=(3,0)\begin{aligned} f &= x^2-6x\\ f_x &= 2x - 6 &= 0 \qquad \implies (x,y)=(3,0) \end{aligned} For x=4x=4: f=(4)2+2y2−6(4)−8y=2y2−8y−8fy=4y−8=0⟹(x,y)=(4,2)\begin{aligned} f &= (4)^2+2y^2-6(4)-8y = 2y^2-8y-8\\ f_y &= 4y-8 = 0 \qquad \implies (x,y)=(4,2) \end{aligned} For y=3y=3: f=x2+2(3)2−6x−8(3)=x2−6x−6fx=2x−6=0⟹(x,y)=(3,3)\begin{aligned} f &= x^2+2(3)^2-6x-8(3) = x^2-6x-6\\ f_x &= 2x-6 = 0 \qquad \implies (x,y)=(3,3) \end{aligned} We make a table of values for each critical point we found, along with each vertex.

    Region xx yy ff Classification
    Interior 33 22 −17-17 Minimum
    Edge 00 22 −8-8
    Edge 33 00 −9-9
    Edge 44 22 −16-16
    Edge 33 33 −15-15
    Vertex 00 00 00 Maximum
    Vertex 44 00 −8-8
    Vertex 00 33 −6-6
    Vertex 44 33 −14-14

    So from the table we can conclude that the minimum of the function is f=−17f=-17 at the interior point (3,2)(3,2), and the maximum is f=0f=0 at the vertex (0,0)(0,0).

    wll 

    [×]
  38. Find the locations and values of the global maximum and minimum of the function f(x,y)=sin⁡(x)cos⁡(y)f(x,y)=\sin(x)\cos(y) inside or on the square π4≤x≤3π4\dfrac{\pi}{4} \le x \le \dfrac{3\pi}{4} and −π4≤y≤π4-\,\dfrac{\pi}{4} \le y \le \dfrac{\pi}{4}.

    Hint

    Find the critical points in the interior, the edges, and the vertices. Then compare the values at those points to determine the minimum and maximum of the function. When solving for critical points, be sure xx and yy are within the region.

    [×]

    Answer

    The minimum is f=0.5f=0.5 at all four verticies, and the maximum is f=1f=1 at the interior point (π2,0)\left(\dfrac{\pi}{2},0\right).

    [×]

    Solution

    To find the critical points inside the square, we set the partial derivatives of f=sin⁡(x)cos⁡(y)f=\sin(x)\cos(y) equal to 00 and solve. Notice that cos⁡(y)≠0\cos(y) \neq 0 on −π4≤y≤π4-\,\dfrac{\pi}{4} \le y \le \dfrac{\pi}{4}, and sin⁡(x)≠0\sin(x) \neq 0 on π4≤x≤3π4\dfrac{\pi}{4} \le x \le \dfrac{3\pi}{4}. fx=cos⁡(x)cos⁡(y)=0⟹x=π2fy=−sin⁡(x)sin⁡(y)=0⟹y=0\begin{aligned} f_x = \cos(x)\cos(y) = 0 \qquad \implies x = \dfrac{\pi}{2}\\ f_y = -\sin(x)\sin(y) = 0 \qquad \implies y = 0 \end{aligned} The critical point (π2,0)\left(\dfrac{\pi}{2},0\right) lies within the square.

    Next, we find all the critical points along the edges of the square, represented by x=π4x=\dfrac{\pi}{4}, x=3π4x=\dfrac{3\pi}{4}, y=−π4y=-\,\dfrac{\pi}{4}, and y=π4y=\dfrac{\pi}{4}. We evaluate the function on each edge and then equate the derivative to zero and solve.
    For x=π4x=\dfrac{\pi}{4}: f=22cos⁡(y)fy=−22sin⁡(y)=0⟹(x,y)=(π4,0)\begin{aligned} f &= \dfrac{\sqrt{2}}{2}\cos(y)\\ f_y &= -\,\dfrac{\sqrt{2}}{2}\sin(y) &= 0 \qquad \implies (x,y)=\left(\dfrac{\pi}{4},0\right) \end{aligned} For x=3π4x=\dfrac{3\pi}{4}: f=22cos⁡(y)fy=−22sin⁡(y)=0⟹(x,y)=(3π4,0)\begin{aligned} f &= \dfrac{\sqrt{2}}{2}\cos(y)\\ f_y &= -\,\dfrac{\sqrt{2}}{2}\sin(y) &= 0 \qquad \implies (x,y)=\left(\dfrac{3\pi}{4},0\right) \end{aligned} For y=−π4y=-\,\dfrac{\pi}{4}: f=22sin⁡(x)fx=22cos⁡(x)=0⟹(x,y)=(π2,−π4)\begin{aligned} f &= \dfrac{\sqrt{2}}{2}\sin(x)\\ f_x &= \dfrac{\sqrt{2}}{2}\cos(x) = 0 \qquad \implies (x,y)=\left(\dfrac{\pi}{2},-\,\dfrac{\pi}{4}\right) \end{aligned} For y=π4y=\dfrac{\pi}{4}: f=22sin⁡(x)fx=22cos⁡(x)=0⟹(x,y)=(π2,π4)\begin{aligned} f &= \dfrac{\sqrt{2}}{2}\sin(x)\\ f_x &= \dfrac{\sqrt{2}}{2}\cos(x) = 0 \qquad \implies (x,y)=\left(\dfrac{\pi}{2},\dfrac{\pi}{4}\right) \end{aligned} We make a table of values for each critical point we found, along with each vertex.

    Region xx yy ff Classification
    Interior π2\dfrac{\pi}{2} 00 11 Maximum
    Edge π4\dfrac{\pi}{4} 00 22≈0.707\dfrac{\sqrt{2}}{2} \approx 0.707
    Edge 3π4\dfrac{3\pi}{4} 00 22≈0.707\dfrac{\sqrt{2}}{2} \approx 0.707
    Edge π2\dfrac{\pi}{2} −π4-\,\dfrac{\pi}{4} 22≈0.707\dfrac{\sqrt{2}}{2} \approx 0.707
    Edge π2\dfrac{\pi}{2} π4\,\dfrac{\pi}{4} 22≈0.707\dfrac{\sqrt{2}}{2} \approx 0.707
    Vertex π4\dfrac{\pi}{4} −π4-\,\dfrac{\pi}{4} 0.50.5 Minimum
    Vertex π4\dfrac{\pi}{4} π4\dfrac{\pi}{4} 0.50.5 Minimum
    Vertex 3π4\dfrac{3\pi}{4} −π4-\,\dfrac{\pi}{4} 0.50.5 Minimum
    Vertex 3π4\dfrac{3\pi}{4} π4\dfrac{\pi}{4} 0.50.5 Minimum

    So from the table we can conclude that the minimum of the function is f=0.5f=0.5 at the verticies, and the maximum is f=1f=1 at (π2,0)\left(\dfrac{\pi}{2},0\right) in the interior.

    wll 

    [×]
  39. Find the locations and values of the global maximum and minimum of the function f(x,y)=3x+4yf(x,y)=3x+4y inside or on the quarter of the circle x2+y2=25x^2+y^2=25 in the 1st1^\text{st} quadrant.

    Hint

    Find the critical points in the interior, the edges, and the verticies. Then compare the values at those points to determine the minimum and maximum of the function. Use a combination of Lagrange multipliers and contraint elimination as convenient.

    [×]

    Answer

    The minimum of the function in the quarter circle is f=0f=0 at (0,0)(0,0), and the maximum is f=25f=25 at (3,4)(3,4) along the arc.

    [×]

    Solution

    To find the critical points inside the quarter circle, we set the partial derivatives of f=3x+4yf=3x+4y equal to 00 and solve. fx=3≠0fy=4≠0\begin{aligned} f_x = 3 \neq 0 \qquad \\ f_y = 4 \neq 0 \qquad \end{aligned} Since there are no solutions, there are no critical points within the quarter circle.

    Next, we find all the critical points along the straight edges of the quarter circle, represented by x=0x=0 or y=0y=0. We evaluate the function on each edge and set the derivatve equal to zero.
    For x=0x=0: f=4yfy=4≠0\begin{aligned} f &= 4y\\ f_y &= 4 \neq 0 \end{aligned} For y=0y=0: f=3xfx=3≠0\begin{aligned} f &= 3x\\ f_x &= 3 \neq 0 \end{aligned} Again this implies that there is no critical point on y=0y=0 or x=0x=0

    Finally, we look for critical points along the arc of the circle. This will be the point where the level set of f=3x+4yf=3x+4y is tangent to the circle, as shown in the plot. This is best found using Lagrange multipliers. The gradients of the functions are: ∇⃗f=(3,4)∇⃗g=(2x,2y)\begin{aligned} \vec\nabla f &=\left(3,4\right) \\ \vec\nabla g &=\left(2x,2y\right) \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are

    x_lines_quarter_circ

    3=λ2x(1)4=λ2y(2)\begin{aligned} 3 = \lambda 2x \qquad \text{(1)}\\ 4 = \lambda 2y \qquad \text{(2)} \end{aligned} We divide equation (1) by equation (2) to get 34=xy\dfrac{3}{4}=\dfrac{x}{y} which implies x=34yx=\dfrac{3}{4}y. We plug this into the contraint g=x2+y2=25g=x^2+y^2=25 to get: (3y4)2+y2=25y2=16y=±4\begin{aligned} \left(\dfrac{3y}{4}\right)^2+y^2&=25 \\ y^2&=16 \\ y&=\pm 4 \end{aligned} We use the positive solution, y=4y=4, to solve for xx because our arc is in the first quadrant. x=34⋅4=3\begin{aligned} x=\dfrac{3}{4}\cdot 4 =3 \end{aligned} So our only critical point is (3,4)(3, 4). We make a table of values of f=3x+4yf=3x+4y for each critical point and each vertex:

    Region xx yy ff Classification
    Edge 33 44 2525 Maximum
    Vertex 00 00 00 Minimum
    Vertex 00 55 2020
    Vertex 55 00 1515

    So from the table we can conclude that the minimum of the function in the quarter circle is f=0f=0 at (0,0)(0,0), and the maximum is f=25f=25 at (3,4)(3,4) along the arc.

    wll 

    [×]
  40. Find the locations and values of the global maximum and minimum of the function f(x,y,z)=2x+3y+4zf(x,y,z)=2x+3y+4z inside or on the eighth of the sphere x2+y2+z2=29x^2+y^2+z^2=29 in the 1st1^\text{st} octant.

    x_eighth_sphere_planes_prob

    Hint

    Find the critical points for each volume, surface, and edge. Then compare the values of the functions at those points and the vertices to determine the minimum and maximum. Lagrange multipliers may be useful.

    [×]

    Answer

    The minimum of the function in the object is f=0f=0 at (0,0,0)(0,0,0), and the maximum is f=29f=29 at (2,3,4)(2,3,4).

    [×]

    Solution

    We begin by finding critical points within the object (1st octant of a sphere centered at the origin). We set the partial derivatives of f=2x+3y+4zf=2x+3y+4z equal to 00 and solve. fx=2≠0fy=3≠0fz=4≠0\begin{aligned} f_x = 2 \neq 0 \\ f_y = 3 \neq 0 \\ f_z = 4 \neq 0 \end{aligned} Since the derivatives never equal zero, there are no critical points inside the object.

    Next, we find all the critical points along the flat surfaces of the object, represented by x=0x=0, y=0y=0, and z=0z=0. We evaluate the function on each surface and set the derivatve equal to zero. f=3y+4zf=2x+4zf=2x+3yfy=3≠0fx=2≠0fx=2≠0fz=4≠0fz=4≠0fy=3≠0\begin{aligned} f &= 3y+4z \qquad& f &= 2x+4z \qquad& f &= 2x+3y\\ f_y &= 3 \neq 0 \qquad& f_x &= 2 \neq 0 \qquad& f_x &= 2 \neq 0\\ f_z &= 4 \neq 0 \qquad& f_z &= 4 \neq 0 \qquad& f_y &= 3 \neq 0 \end{aligned} This implies that there is no critical point on any of the flat surfaces.

    Next, we can use Lagrange multipliers with the contraint surface g=x2+y2+z2=29g=x^2+y^2+z^2=29 to find critical points along the spherical surface of the object. The gradients of the functions are: ∇⃗f=⟨2,3,4⟩∇⃗g=⟨2x,2y,2z⟩\begin{aligned} \vec\nabla f &=\langle 2,3,4 \rangle \\ \vec\nabla g &=\langle 2x,2y, 2z\rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g\vec\nabla f=\lambda\vec\nabla g, are 2=λ2x(1)3=λ2y(2)4=λ2z(3)\begin{aligned} 2 = \lambda 2x \qquad \text{(1)}\\ 3 = \lambda 2y \qquad \text{(2)}\\ 4 = \lambda 2z \qquad \text{(3)}\\ \end{aligned} We divide equations (1) and (2) by equation (3) and solve in terms of zz. 24=xz⟹x=z2(5)34=yz⟹y=3z4(6)\begin{aligned} \dfrac{2}{4}=\dfrac{x}{z} &\implies x = \dfrac{z}{2} \qquad& \text{(5)}\\ \dfrac{3}{4}=\dfrac{y}{z} &\implies y = \dfrac{3z}{4} \qquad& \text{(6)} \end{aligned} We plug these into the contraint g=x2+y2+z2=29g=x^2+y^2+z^2=29 (z2)2+(3z4)2+z2=29z2=16z=±4\begin{aligned} \left(\dfrac{z}{2}\right)^2+\left(\dfrac{3z}{4}\right)^2+z^2&=29 \\ z^2&=16 \\ z&=\pm 4 \end{aligned} We use the positive solution, z=4z=4, to solve for xx and yy because our object is in the first octant. x=12⋅4=2y=34⋅4=3\begin{aligned} x&=\dfrac{1}{2}\cdot 4&=2\\ y&=\dfrac{3}{4}\cdot 4&=3 \end{aligned} So the critical point on the spherical surface is (2,3,4)(2, 3, 4).

    Next, we must find the critical points along the edges of the object. The straight edges are represented by (x,0,0)(x,0,0), (0,y,0)(0,y,0), and (0,0,z)(0,0,z). We evaluate the function at these lines and set the derivatives equal to zero. f=2xf=3yf=4zfx=2≠0fy=3≠0fz=4≠0\begin{aligned} f &= 2x \qquad& f &= 3y \qquad& f &= 4z\\ f_x &= 2 \neq 0 \qquad& f_y &= 3 \neq 0 \qquad& f_z &= 4 \neq 0 \end{aligned} Since the derivatives never equal zero, there are no critical points along the straight edges.

    Next, we use Lagrange multipliers to find the critical points along the curved edges.
    For x=0x=0, f=3y+4zf=3y+4z on g=y2+z2=29g=y^2+z^2=29. The Lagrange equations say: 3=λ2y4=λ2z⟹y=3z4 3=\lambda2y \qquad 4=\lambda2z \implies y=\dfrac{3z}{4} So the constraint says: (3z4)2+z2=29⟹z=4295(x,y,z)=(0,3295,4295)\begin{aligned} \left(\dfrac{3z}{4}\right)^2+z^2&=29 \implies z = \dfrac{4\sqrt{29}}{5} \\ (x, y, z) &= \left(0, \dfrac{3\sqrt{29}}{5},\dfrac{4\sqrt{29}}{5}\right) \end{aligned} For y=0y=0, f=2x+4zf=2x+4z on g=x2+z2=29g=x^2+z^2=29. The Lagrange equations say: 2=λ2x4=λ2z⟹x=z2 2=\lambda2x \qquad 4=\lambda2z \implies x=\dfrac{z}{2} So the constraint says: (z2)2+z2=29⟹z=2295(x,y,z)=(295,0,2295)\begin{aligned} \left(\dfrac{z}{2}\right)^2+z^2&=29 \implies z = 2 \sqrt{\dfrac{29}{5}} \\ (x, y, z) &= \left(\sqrt{\dfrac{29}{5}}, 0,2 \sqrt{\dfrac{29}{5}}\right) \end{aligned} For z=0z=0, f=2x+3yf=2x+3y on g=x2+y2=29g=x^2+y^2=29. The Lagrange equations say: 2=λ2x3=λ2y⟹y=3x2 2=\lambda2x \qquad 3=\lambda2y \implies y=\dfrac{3x}{2} So the constraint says: x2+(3x2)2=29⟹x=22913(x,y,z)=(22913,32913,0)\begin{aligned} x^2+\left(\dfrac{3x}{2}\right)^2&=29 \implies x = 2 \sqrt{\dfrac{29}{13}} \\ (x, y, z) &= \left(2 \sqrt{\dfrac{29}{13}}, 3 \sqrt{\dfrac{29}{13}},0\right) \end{aligned} We make a table of values of f=2x+3y+4zf=2x+3y+4z for each critical point we found, along with each vertex.

    Region xx yy zz ff Classification
    Surface 22 33 44 2929 Maximum
    Edge 00 3295\dfrac{3\sqrt{29}}{5} 4295\dfrac{4\sqrt{29}}{5} 529≈26.95\sqrt{29}\approx 26.9
    Edge 295\sqrt{\dfrac{29}{5}} 00 22952\sqrt{\dfrac{29}{5}} 2145≈24.12\sqrt{145}\approx 24.1
    Edge 229132 \sqrt{\dfrac{29}{13}} 329133 \sqrt{\dfrac{29}{13}} 00 377≈19.4\sqrt{377}\approx 19.4
    Vertex 29\sqrt{29} 00 00 229≈10.82\sqrt{29}\approx 10.8
    Vertex 00 29\sqrt{29} 00 329≈16.23\sqrt{29}\approx 16.2
    Vertex 00 00 29\sqrt{29} 429≈21.54\sqrt{29}\approx 21.5
    Vertex 00 00 00 00 Minimum

    So from the table, we can conclude that the minimum of the function is f=0f=0 at the vertex (0,0,0)(0,0,0), and the maximum is f=29f=29 at the point (2,3,4)(2,3,4) on the spherical surface. In the plot, the level sets of ff are the planes in blue or orange. The upper orange level set is for f=29f=29 which is tangent to the sphere at (2,3,4)(2,3,4). The lower orange level set is for f=0f=0 which passes through the point (0,0,0)(0,0,0). All the points in the interior have values of ff between 00 and 2929.

    x_eighth_sphere_planes_sol

    wll,tj 

    [×]
  41. Consider the function f(x,y)=2x2−4xy−2x+4y2−4yf(x,y)=2x^2-4xy-2x+4y^2-4y and the square −4≤x≤6-4 \le x \le 6 and −1≤y≤9-1 \le y \le 9.

    In the plot, the square is red and the level sets of ff are blue.

    x_ellipses_square_prob
    1. Find the locations and values of the global maximum and minimum of the function ff on the (edges of the) square.

      Hint

      Find the critical points on each edges. Then make a table of the values of ff at those points and the vertices to determine the minimum and maximum locations. When solving for critical points, be sure xx and yy are within the region.

      [×]

      Answer

      Global Maximum: f=472at the vertex(x,y)=(−4,9) f=472 \quad \text{at the vertex} \quad (x,y)=(-4,9) Global Minimum: f=7.5at the edge point(x,y)=(−12,−1) f=7.5 \quad \text{at the edge point} \quad (x,y)=\left(-\,\dfrac{1}{2},-1\right)

      [×]

      Solution

      We find the critical points along all the edges of the square, represented by x=−4x=-4, x=6x=6, y=−1y=-1, and y=9y=9. We evaluate the function on each edge and then equate the derivative to zero and solve.
      For x=−4x=-4: f=2(−4)2−4(−4)y−2(−4)+4y2−4y=4y2+12y+40fy=8y+12=0⟹(x,y)=(−4,−32)\begin{aligned} f &= 2(-4)^2-4(-4)y-2(-4)+4y^2-4y = 4y^2 +12y +40\\ f_y &= 8y + 12= 0 \qquad \implies (x,y)=\left(-4,\dfrac{-3}{2}\right) \end{aligned} This is not on the edge of the square because y=−32y=\dfrac{-3}{2} does not satisfy −1≤y≤9-1 \le y \le 9.
      For x=6x=6: f=2(6)2−4(6)y−2(6)+4y2−4y=4y2−28y+60fy=8y−28=0⟹(x,y)=(6,72)\begin{aligned} f &= 2(6)^2-4(6)y-2(6)+4y^2-4y = 4y^2 -28y +60\\ f_y &= 8y - 28 = 0 \qquad \implies (x,y)=\left(6,\dfrac{7}{2}\right) \end{aligned} For y=−1y=-1: f=2x2−4x(−1)−2x+4(−1)2−4(−1)=2x2+2x+8fx=4x+2=0⟹(x,y)=(−12,−1)\begin{aligned} f &= 2x^2-4x(-1)-2x+4(-1)^2-4(-1) = 2x^2 +2x + 8\\ f_x &= 4x+ 2= 0 \qquad \implies (x,y)=\left(\dfrac{-1}{2},-1\right) \end{aligned} For y=9y=9: f=2x2−4x(9)−2x+4(9)2−4(9)=2x2−38x+288fx=4x−38=0⟹(x,y)=(192,9)\begin{aligned} f &= 2x^2-4x(9)-2x+4(9)^2-4(9) = 2x^2 -38x +288\\ f_x &= 4x -38= 0 \qquad \implies (x,y)=\left(\dfrac{19}{2},9\right) \end{aligned} This is not on the edge of the square because x=−192x=\dfrac{-19}{2} does not satisfy −4≤x≤6-4 \le x \le 6.

      We make a table of values of f=2x2−4xy−2x+4y2−4yf=2x^2-4xy-2x+4y^2-4y for each critical point we found, along with each vertex:

      Region xx yy ff Classification
      Edge 66 72\dfrac{7}{2} 1111
      Edge −12\dfrac{-1}{2} −1-1 7.57.5 Minimum
      Vertex −4-4 −1-1 3232
      Vertex −4-4 99 472472 Maximum
      Vertex 66 −1-1 9292
      Vertex 66 99 132132

      So from the table we conclude that the minimum of the function in the square is f=7.5f=7.5 at the edge point (−12,−1)\left(-\,\dfrac{1}{2},-1\right), and the maximum is f=472f=472 at the vertex (−4,9)(-4,9).

      In the animation, the square is red,
      and the level sets of ff are blue.
      The level sets for the minimum and maximum on the square flash in orange.

      x_ellipses_on_square_sol

      wll,tj 

      [×]
    2. Find the locations and values of the global maximum and minimum of the function ff inside and on the square.

      Hint

      Find the critical points in the interior and the edges. Then make a table of the values of ff at those points and the vertices to determine the minimum and maximum locations. When solving for critical points, be sure xx and yy are within the region.

      [×]

      Answer

      Global Maximum: f=472at the interior point(x,y)=(−4,9) f=472 \quad \text{at the interior point} \quad (x,y)=(-4,9) Global Minimum: f=−5at the vertex(x,y)=(2,32) f=-5 \quad \text{at the vertex} \quad (x,y)=\left(2, \dfrac{3}{2}\right)

      [×]

      Solution

      To find the critical points inside the square, we set the partial derivatives of f=2x2−4xy−2x+4y2−4yf=2x^2-4xy-2x+4y^2-4y equal to 00 and solve. fx=4x−4y−2=0fy=−4x+8y−4=0\begin{aligned} f_x &= 4x-4y-2 &= 0 \\ f_y &= -4x+8y-4 &= 0 \end{aligned} We add the two equations to eliminate xx terms, producing the equation 4y−6=04y-6=0. This implies that our critical point is (x,y)=(2,32)(x,y)=\left(2, \dfrac{3}{2}\right) which we observe is inside the square.

      We add this to the table of values we found in part (a):

      Region xx yy ff Classification
      Interior 22 32\dfrac{3}{2} −5-5 Minimum
      Edge 66 72\dfrac{7}{2} 1111
      Edge −12\dfrac{-1}{2} −1-1 7.57.5
      Vertex −4-4 −1-1 3232
      Vertex −4-4 99 472472 Maximum
      Vertex 66 −1-1 9292
      Vertex 66 99 132132

      So from the table we can conclude that the minimum of the function in or on the square is f=−5f=-5 at the interior point (2,32)\left(2, \dfrac{3}{2}\right), and the maximum is f=472f=472 at the vertex (−4,9)(-4,9).

      In the animation, the square is red,
      and the level sets of ff are blue.
      The level sets for the minimum and maximum in or on the square flash in orange.
      The level set which just touches the square also flashes orange, but is no longer the minimum.

      x_ellipses_inon_square_sol

      wll,tj 

      [×]
  42. The plane x+2y+3z=1x+2y+3z =1 intersects the cylinder x2+y2=1x^2+y^2=1 in an ellipse. Find the points on this ellipse closest and farthest from the origin.

    1. Solve using Lagrange multipliers.

    2. Solve using a paramaterization.

    x_ell_plane_cyl
    1

    Hint

    Simplify your calculations by optimizing the squared distance. Find the gradients of the distance squared and the two contraints. Then, solve the Lagrange equations, ∇⃗f=λ∇⃗g+μ∇⃗h\vec\nabla f=\lambda\vec\nabla g + \mu\vec\nabla h, along with the constraints to find critical points. Finally, make a table of values to determine the distances of the critical points found.

    [×]

    Answer

    The points closest to the origin is (1,0,0)(1,0,0) and (−35,45,0)\left(-\dfrac{3}{5},\dfrac{4}{5},0\right). The point farthest from the origin is (−15,−25,1+53)\left(- \dfrac{1}{\sqrt{5}}, - \dfrac{2}{\sqrt{5}}, \dfrac{1 + \sqrt{5}}{3}\right).

    [×]

    Solution

    We minimize the square of the distance function from the origin using f=D2=x2+y2+z2f=D^2 = x^2+y^2+z^2 on the intersection of the plane g=x+2y+3z=1g=x+2y+3z =1 and the cylinder h=x2+y2=1h=x^2+y^2=1 The gradients of the functions are ∇⃗f=⟨2x,2y,2z⟩∇⃗g=⟨1,2,3⟩∇⃗h=⟨2x,2y,0⟩\begin{aligned} \vec\nabla f&=\langle 2x, 2y, 2z \rangle \\ \vec\nabla g&=\langle 1,2,3 \rangle \\ \vec\nabla h&=\langle 2x,2y,0 \rangle \end{aligned} The Lagrange equations, ∇⃗f=λ∇⃗g+μ∇⃗h\vec\nabla f=\lambda\vec\nabla g + \mu\vec\nabla h, are 2x=λ+μ2x(1)2y=λ2+μ2y(2)2z=λ3(3)\begin{aligned} 2x&=\lambda + \mu 2x \qquad& \text{(1)} \\ 2y&=\lambda 2 + \mu 2y \qquad& \text{(2)} \\ 2z&=\lambda 3 \qquad& \text{(3)} \end{aligned} Equation (3) implies λ=23z\lambda=\dfrac{2}{3}z and we eliminate λ\lambda by plugging it into equations (1) and (2). 2x=23z+μ2x(4)2y=43z+μ2y(5)\begin{aligned} 2x&=\dfrac{2}{3}z + \mu 2x \qquad& \text{(4)} \\ 2y&=\dfrac{4}{3}z + \mu 2y \qquad& \text{(5)} \end{aligned} To eliminate μ\mu, we multiply equation (4) by yy and equation (5) by xx and subtract. This yields: 0=(23y−43x)z 0=\left(\dfrac{2}{3}y-\dfrac{4}{3}x\right)z There are two scenarios, where z=0z=0, or (23y−43x)=0\left(\dfrac{2}{3}y-\dfrac{4}{3}x\right)=0. We use both constraints to find critical points.
    Case 1: z=0z=0 x+2y=1x2+y2=1⟹x=1−2y(1−2y)2+y2=1⟹x=1−2y−4y+5y2=0\begin{aligned} \begin{array}{r} x+2y=1 \\ x^2+y^2=1 \end{array} \implies \begin{array}{l} x=1-2y \\ (1-2y)^2+y^2=1 \end{array} \implies \begin{array}{l} x=1-2y \\ -4y+5y^2=0 \end{array} \end{aligned} The last equation says y=0y = 0 or y=45y=\dfrac{4}{5}. Then x=1x=1 or x=−35x=-\dfrac{3}{5} respectively. So the two critical points are: (x,y,z)=(1,0,0)and(−35,45,0) (x,y,z) = (1,0,0) \qquad \text{and} \qquad \left(-\dfrac{3}{5},\dfrac{4}{5},0\right) Case 2: 23y−43x=0\dfrac{2}{3}y-\dfrac{4}{3}x=0 or equivalently y=2xy = 2x 5x+3z=15x2=1⟹x=±15andy=±253z=1−5(±15)orz=1∓53\begin{aligned} \begin{array}{r} 5x+3z=1& \\ 5x^2=1& \end{array} \implies \begin{array}{l} x = \pm \dfrac{1}{\sqrt{5}} \quad \text{and} \quad y = \pm \dfrac{2}{\sqrt{5}} \\ 3z=1-5\left(\pm \dfrac{1}{\sqrt{5}}\right)\quad \text{or} \quad z=\dfrac{1 \mp \sqrt{5}}{3} \end{array} \end{aligned} So the two critical points are: (x,y,z)=(15,25,1−53)and(−15,−25,1+53) (x,y,z) = (\dfrac{1}{\sqrt{5}}, \dfrac{2}{\sqrt{5}}, \dfrac{1-\sqrt{5}}{3}) \qquad \text{and} \qquad \left(-\,\dfrac{1}{\sqrt{5}},-\,\dfrac{2}{\sqrt{5}}, \dfrac{1+\sqrt{5}}{3}\right) We make a table of values to determine the distance of the critical points.

    xx yy zz\quad D2D^2 Classification
    11 00 00\quad 11 Minimum
    −35-\dfrac{3}{5} 45\dfrac{4}{5} 00\quad 11 Minimum
    15\dfrac{1}{\sqrt{5}} 25\dfrac{2}{\sqrt{5}} 1−53\dfrac{1 - \sqrt{5}}{3}\quad 15−259≈1.17\dfrac{15 - 2\sqrt{5}}{9}\approx 1.17
    −15-\dfrac{1}{\sqrt{5}} −25-\dfrac{2}{\sqrt{5}} 1+53\dfrac{1 + \sqrt{5}}{3}\quad 15+259≈2.16\dfrac{15 + 2\sqrt{5}}{9}\approx 2.16 Maximum

    So from the table, we conclude that the points closest to the origin is (1,0,0)(1,0,0) and (−35,45,0)\left(-\dfrac{3}{5},\dfrac{4}{5},0\right). The point farthest from the origin is (−15,−25,1+53)\left(- \dfrac{1}{\sqrt{5}}, - \dfrac{2}{\sqrt{5}}, \dfrac{1 + \sqrt{5}}{3}\right).

    wll,tj 

    [×]

    Hint

    Simplify your calculations by optimizing the squared distance. Start by paramaterizing the cylinder in the xx-yy plane.

    [×]

    Answer

    The points closest to the origin is The point farthest from the origin is

    [×]

    Solution

    Using cyclindrical coordinates, we can paramaterize f=D2f=D^2 on the ellipse. Since the raidus of the cylinder is r=1r=1: x=cos⁡θy=sin⁡θ\begin{aligned} x = \cos\theta \\ y = \sin\theta \end{aligned} We use the plane equation to solve for zz and paramaterize: z=1−x−2y3z=1−cos⁡θ−2sin⁡θ3\begin{aligned} z&=\dfrac{1-x-2y}{3} \\ z&=\dfrac{1-\cos\theta-2\sin\theta}{3} \end{aligned} Substituting these values into f=D2=x2+y2+z2f=D^2=x^2+y^2+z^2: f=cos⁡2θ+sin⁡2θ+(1−cos⁡θ−2sin⁡θ3)2=1+(1−cos⁡θ−2sin⁡θ3)2\begin{aligned} f &= \cos^2\theta + \sin^2\theta + \left(\dfrac{1-\cos\theta-2\sin\theta}{3}\right)^2 \\ &= 1 + \left(\dfrac{1-\cos\theta-2\sin\theta}{3}\right)^2 \end{aligned} To find the critical points We differentiate ff and equate it to zero: f′=2(1−cos⁡θ−2sin⁡θ3)⋅(sin⁡θ−2cos⁡θ3)=0(1−cos⁡θ−2sin⁡θ)⋅(sin⁡θ−2cos⁡θ)=0\begin{aligned} f' = 2\left(\dfrac{1-\cos\theta-2\sin\theta}{3}\right) \cdot \left(\dfrac{\sin\theta-2\cos\theta}{3}\right) &= 0 \\ (1-\cos\theta-2\sin\theta) \cdot (\sin\theta-2\cos\theta) &= 0 \end{aligned} Case 1: 1−cos⁡θ−2sin⁡θ=01-\cos\theta-2\sin\theta=0 1−cos⁡θ−2sin⁡θ=01−cos⁡θ=2sin⁡θ(1−cos⁡θ)2=4sin⁡2θ1−2cos⁡θ+cos⁡2θ=4(1−cos⁡2θ)5cos⁡2θ−2cos⁡θ−3=0(5cos⁡θ+3)(cos⁡θ−1)=0cos⁡θ=1,−35θ=0,arccos⁡−35\begin{aligned} 1-\cos\theta-2\sin\theta &= 0 \\ 1 - \cos\theta &= 2\sin\theta \\ (1 - \cos\theta)^2 &= 4\sin^2\theta \\ 1 - 2\cos\theta + \cos^2\theta &= 4(1-\cos^2\theta) \\ 5\cos^2\theta - 2\cos\theta - 3 &= 0 \\ (5\cos\theta+3)(\cos\theta-1) &= 0 \\ \cos\theta &= 1, \dfrac{-3}{5} \\ \theta &= 0, \arccos{\dfrac{-3}{5}} \end{aligned} Case 2: sin⁡θ−2cos⁡θ=0\sin\theta-2\cos\theta=0 sin⁡θ−2cos⁡θ=0sin⁡θ=2cos⁡θtan⁡θ=2θ=arctan⁡(2),arctan⁡(2)+π\begin{aligned} \sin\theta-2\cos\theta &= 0 \\ \sin\theta &= 2\cos\theta \\ \tan\theta &= 2 \\ \theta &= \arctan(2), \arctan(2) + \pi \end{aligned} We make a table of values of D2=1+(1−cos⁡θ−2sin⁡θ3)2D^2=1+\left(\dfrac{1-\cos\theta-2\sin\theta}{3}\right)^2 to determine the distance of the critical points.

    θ\theta D2D^2 Classification
    00 11 Minimum
    arccos⁡−35\arccos{\dfrac{-3}{5}} 11 Minimum
    arctan⁡(2)\arctan(2) 1+(1−53)2≈1.171 + \left(\dfrac{1-\sqrt{5}}{3}\right)^2 \approx 1.17
    arctan⁡(2)+π\arctan(2) + \pi 1+(1+53)2≈2.161 + \left(\dfrac{1+\sqrt{5}}{3}\right)^2 \approx 2.16 Maximum

    So from the table, we conclude that the points closest to the origin is (1,0,0)(1,0,0) and (−35,45,0)\left(-\dfrac{3}{5},\dfrac{4}{5},0\right). The point farthest from the origin is (−15,−25,1+53)\left(- \dfrac{1}{\sqrt{5}}, - \dfrac{2}{\sqrt{5}}, \dfrac{1 + \sqrt{5}}{3}\right).

    wll,tj 

    [×]
  43. Maximize the function f=x2−y2+2zf=x^2-y^2+2z on the line of intersection of the planes y+z=0y+z=0 and z=2xz=2x.

    Hint

    Determine the geometry of the constraints and attempt to parameterize the intersection. Maximize the function with respect to the parameterized variable.

    [×]

    Answer

    The maximum value of ff at the intersection of the planes y+z=0y+z=0 and z=2xz=2x is 43\dfrac{4}{3}.

    [×]

    Solution

    To maximize the function f=x2−y2+2zf=x^2-y^2+2z, we parameterize the function along the line of intersection of the planes y+z=0y+z=0 and z=2xz=2x, as follows: x=tz=2x=2ty=−z=−2t\begin{aligned} x&=t \\ z &= 2x = 2t \\ y &=-z=-2t \end{aligned} We plug the parameterized variables into ff: f=t2−(−2t)2+2(2t)=−3t2+4t\begin{aligned} f&=t^2-(-2t)^2+2(2t) \\ &= -3t^2+4t \end{aligned} To find the critical points along this intersection, we take the derivatve of ff with respect to tt and equate it to zero: f′=−6t+4=0⟹t=23 f'=-6t + 4=0 \implies t = \dfrac{2}{3} To determine if this is a minimum or maximum, we use the Second Derivative Test. f′′=−6<0 f''=-6 \lt 0 Since f′′f'' is always less than zero, the critical point at t=23t=\dfrac{2}{3} is a maximum. We use tt to get cartesian coordinates and the maximum value of ff. (x,y,z)=(23,−43,43)f=−3(23)2+4(23)=43\begin{aligned} (x,y,z) &= \left(\dfrac{2}{3}, \dfrac{-4}{3},\dfrac{4}{3}\right) \\ f &= -3\left(\dfrac{2}{3}\right)^2+4\left(\dfrac{2}{3}\right) = \dfrac{4}{3} \end{aligned} So the maximum value of ff at the intersection of the planes y+z=0y+z=0 and z=2xz=2x is 43\dfrac{4}{3}.

    wll 

    [×]
  44. A box manufacturer wants to make a rectangular solid cardboard box to contain 4000cm34000\,\text{cm}^3. The box is folded from a single rectangular piece of cardboard so that the flaps from 22 sides just meet at the centerline of the top and bottom. The flaps from the other 22 sides do not need to reach the centerline of the top and bottom. What are the dimensions of the original rectangle of cardboard used to make the box which uses the least amount of cardboard?

    x_box_uneven

    Hint

    Let LL and WW be the length and width of the original piece of cardboard. Let xx, yy and zz be the length, width and height of the final box. Write out the constraint for the volume. Write out the constraints relating the original cardboard dimensions to the final box dimensions. What function are you minimizing? How many variables are there? How many constraints?

    [×]

    Answer

    The dimensions of the original carboard piece that minimize the amount of cardboard used are: L=60cmL=60\,\text{cm} and W=30cmW=30\,\text{cm}.

    [×]

    Solution

    Let xx, yy and zz be the length, width and height of the final box, resp. Let LLand WW be the length and width of the original piece of cardboard, resp. The length of the orignal cardboard is the perimeter of the xyxy top, so it can be written as L=2x+2yL=2x+2y. Similarly, the width of the orignal cardboard is height of the box plus the two flaps which meet exactly in the middle, so it can be written as W=y2+z+y2=y+zW=\dfrac{y}{2} + z + \dfrac{y}{2} = y + z. Therefore, the amount of cardboard used is: A(z,y,x)=LW=(2x+2y)(y+z)=2xy+2xz+2y2+2yz\begin{aligned} A(z, y, x)&=LW= (2x+2y)(y + z) \\ &=2xy+2xz+2y^2+2yz \end{aligned} And the total enclosed volume is: V(z,y,x)=xyz=4000 V(z,y,x) = xyz = 4000 The gradients of the functions are ∇⃗A=⟨2y+2z,2x+4y+2z,2x+2y⟩∇⃗V=⟨yz,xz,xy⟩\begin{aligned} \vec\nabla A&=\langle 2y+2z,2x+4y+2z,2x+2y \rangle \\ \vec\nabla V&=\langle yz,xz,xy \rangle \end{aligned} The Lagrange equations, ∇⃗A=λ∇⃗V\vec\nabla A=\lambda\vec\nabla V, are 2y+2z=λyz(1)2x+4y+2z=λxz(2)2x+2y=λxy(3)\begin{aligned} 2y+2z&= \lambda yz \qquad \text{(1)} \\ 2x+4y+2z&=\lambda xz \qquad \text{(2)} \\ 2x+2y&= \lambda xy \qquad \text{(3)} \end{aligned} We multiply equation (1) by xx, equation (2) by yy, and equation (3) by zz. We then equate and solve: λzyx=2xy+2xz=2xy+4y2+2yz=2xz+2yz⟹xz=2y2+yzx=z\begin{aligned} \lambda zyx&= 2xy+2xz=2xy+4y^2+2yz=2xz+2yz \\ &\implies \quad xz = 2y^2+yz \qquad x = z \end{aligned} Substituting x=zx=z into the volume constraint gives: V=z2y=4000y=4000z2\begin{aligned} V &=z^2y=4000\\ y &= \dfrac{4000}{z^2} \\ \end{aligned} We substitute this into xz=2y2+yzxz = 2y^2+yz and solve for zz: z(z)=2(4000z2)2+(4000z2)(z)z6−4000z3−2(4000)2=0\begin{aligned} z(z) = 2\left(\dfrac{4000}{z^2}\right)^2+\left(\dfrac{4000}{z^2}\right)&(z) \\ z^6-4000z^3&-2(4000)^2 = 0 \end{aligned} This is a quadratic equation for z3z^3. We use the quadratic formula: z3=4000±40002+4⋅2(4000)22=8000or−4000\begin{aligned} z^3 &= \dfrac{4000 \pm \sqrt{4000^2+4\cdot2(4000)^2}}{2} \\ &= 8000\,\text{or} -4000 \\ \end{aligned} We take the positive solution z3=8000z^3=8000. So z=20cmz=20\,\text{cm}, y=10cmy=10\,\text{cm} and x=20cmx=20\,\text{cm}. So the dimensions of the original carboard piece that minimixes the amount of cardboard used are: L=2x+2y=60cmW=y+z=30cm\begin{aligned} L &= 2x + 2y = 60\,\text{cm} \\ W &= y + z = 30\,\text{cm} \end{aligned}

    wll 

    [×]
  45. Challenge Exercises

    It is recommended that you set up the equations using one of the three methods, but then solve them numerically using a Computer Algebra System like Maple or Mathematics or Sage or MatLab.

  46. A paper cup has the shape of a frustum of a cone with a bottom. Let the smaller bottom radius be rr, the larger top radius be RR and the height be hh. If the cup needs to hold 100πcm3100\pi\,\text{cm}^3 of water, what are the dimensions rr, RR and hh which use the least paper?

    x_papercup

    Hint

    The area of the slanted sides of the cup (frustum) is: Afrustum=π(r+R)(R−r)2+h2 A_\text{frustum}=\pi(r+R)\sqrt{(R-r)^2+h^2} The area of the bottom is Abottom=πr2A_\text{bottom}=\pi r^2. So the total area is: Acup=π(r+R)(R−r)2+h2+πr2 A_\text{cup}=\pi(r+R)\sqrt{(R-r)^2+h^2}+\pi r^2 The volume of the cup (frustum) is V=13π(r2+rR+R2)h V=\dfrac{1}{3}\pi(r^2+rR+R^2)h Add an extra constraint which says the slant height is s2=(R−r)2+h2 s^2=(R-r)^2+h^2 so then the total area is: A=π(r+R)s+πr2 A=\pi(r+R)s+\pi r^2 This will eliminate a lot of square roots from the problem.

    [×]

    Answer

    (r,R,h,s)=(3.10,5.66,5.05,5.66) (r, R, h, s)=(3.10, 5.66, 5.05, 5.66)

    [×]

    Solution

    From the hint, the surface area of the cup is A=π(r+R)s+πr2 A=\pi(r+R)s + \pi r^2 where the slant height ss satisfies s2=(R−r)2+h2s^2=(R-r)^2+h^2. So we want to minimize the area AA subject to the constraint that the volume is constant: V=13π(r2+rR+R2)h=100π V=\dfrac{1}{3}\pi(r^2+rR+R^2)h=100\pi and the constraint on the slant height: G=(R−r)2+h2−s2=0 G=(R-r)^2+h^2-s^2=0 We take the order of the variables in the functions as (r,R,h,s)(r,R,h,s). So the gradients are: ∇⃗A=⟨πs+2πr,πs,0,πr+πR⟩∇⃗V=⟨πh3(2r+R),πh3(2R+r),π3(r2+rR+R2),0⟩∇⃗G=⟨2r−2R,2R−2r,2h,−2s⟩\begin{aligned} \vec\nabla A&=\langle \pi s + 2\pi r, \pi s, 0, \pi r+ \pi R \rangle \\ \vec\nabla V&=\left\langle \dfrac{\pi h}{3} (2r + R), \dfrac{\pi h}{3} (2R+r), \dfrac{\pi}{3}(r^2+rR+R^2), 0 \right\rangle \\ \vec\nabla G&=\left\langle 2r-2R, 2R-2r, 2h, -2s \right\rangle \end{aligned} The Lagrange equations, ∇⃗A=λ∇⃗V+μ∇⃗G\vec\nabla A=\lambda\vec\nabla V + \mu\vec\nabla G, are πs+2πr=λπh3(2r+R)+μ(2r−2R)(1)πs=λπh3(2R+r)+μ(2R−2r)(2)0=λπ3(r2+rR+R2)+μ2h(3)πr+πR=−2μs(4)\begin{aligned} \pi s + 2\pi r &= \lambda \dfrac{\pi h}{3} (2r+R) + \mu (2r-2R) \qquad &\text{(1)} \\ \pi s &= \lambda \dfrac{\pi h}{3} (2R+r) + \mu (2R-2r) \qquad &\text{(2)} \\ 0 &= \lambda \dfrac{\pi}{3}(r^2+rR+R^2) + \mu 2h \qquad &\text{(3)} \\ \pi r+ \pi R &= -2\mu s \qquad &\text{(4)} \end{aligned} From here, we use a CAS system to solve the system of equations. The following python code is used to find a numerical solution:

            
    
          from scipy.optimize import fsolve
          import math as m
    
          def equations(p):
            r, R, h, s, l, u = p
            eq1 = (l * m.pi * h)*(2*r+R)/3 + u*(2*r-2*R) - m.pi *s - 2 * m.pi * r
            eq2 = (l * m.pi * h)*(2*R+r)/3 + u*(2*R-2*r) - m.pi *s
            eq3 = (l * m.pi)*(r**2+r*R+R**2)/3 + 2*u*h
            eq4 = m.pi * r + m.pi * R + 2 * u * s
            eq5 = m.pi * h *(r**2+r*R+R**2)/3 - 100 * m.pi
            eq6 = (R-r)**2 + h**2 -s**2
            return [eq1, eq2, eq3, eq4, eq5, eq6]
    
          print(fsolve(equations, (1, 2, 1, 1, 1, 1)))
    
    
          
        

    We use the initial guess (1,2,1,1,1,1)(1, 2, 1, 1, 1, 1) because it is a reasonable set of values for our optimal frustum. It is important to note that your numerical solution may differ depending on your initial conditions. The output of the code produces the values:

          
    
        (3.10, 5.66, 5.05, 5.66, 0.39, -2.43)
    
          
        

    From this output we conclude (r,R,h,s)=(3.10,5.66,5.05,5.66) (r, R, h, s)=(3.10, 5.66, 5.05, 5.66) as well as Îť=5.05\lambda=5.05 and Îź=5.66\mu=5.66. These are plausible dimensions for the frustum, and thus one optimal solution.

    wll,tj 

    [×]
  47. A trash dumpster has the shape shown in the plot. It is a rectangular solid with a triangular prism cut off the top front where there is a diagonal lid. Let WW be the width across the front. Let LL be the length of the base from back to front. Let HH be the height of the back. Let ℓ\ell be the length of the top from the back to the lid. Let hh be the height of the front from the bottom to the lid. The heavy steel base costs $4040 per square foot. The sides, top and lid are lighter steel which cost $2020 per square foot. Every edge needs to be welded which costs $11 per foot. The hinge for the lid costs $1010 per foot including its welding cost. The volume needs to be 150ft3150\,\text{ft}^3. The lid needs to be 15ft215\,\text{ft}^2. What are the dimensions which minimize the total cost? (Solving the resulting equations may require a computer algebra system.)

    x_trash_dumpster

    Hint

    To eliminate a bunch of square roots, it is helpful to introduce an extra variable which is the slant length of the lid: s=(L−l)2+(H−h)2 s=\sqrt{(L-l)^2+(H-h)^2} along with the extra constraint: s2−(L−l)2−(H−h)2=0 s^2-(L-l)^2-(H-h)^2=0 Find the set of Lagrange equations then use a CAS system to find a numerical solution.

    [×]

    Answer

    (W,L,H,l,h,s)=(4.65,4.84,7.20,2.56,4.92,3.22) (W, L, H, l, h, s)=(4.65, 4.84, 7.20, 2.56, 4.92, 3.22)

    [×]

    Solution

    We construct a function for the total cost of construction. First, at $4040 per square foot, the base costs: Cbase=40WL C_{base} = 40WL At $2020 per square foot, the top, front, back, 22 sides, and lid cost: Cbody=20lW+20hW+20HW+40LH−20(L−l)(H−h)+20W(L−l)2+(H−h)2\begin{aligned} C_{body}&= 20lW + 20hW + 20HW + 40LH \\ &\quad- 20(L-l)(H-h) + 20W\sqrt{(L-l)^2+(H-h)^2} \end{aligned} At $1010 per foot, the hinge costs: Chinge=10W C_{hinge} = 10W At $11 per foot, the welding costs: Cweld=3W+2L+2H+2h+2l C_{weld} = 3W + 2L + 2H + 2h + 2l So the total cost of the dumpster is: C=40WL+20lW+20hW+20HW+40LH−20(L−l)(H−h)+20W(L−l)2+(H−h)2+13W+2L+2H+2h+2l\begin{aligned} C&=40WL + 20lW + 20hW + 20HW + 40LH - 20(L-l)(H-h) \\ &\quad+ 20W\sqrt{(L-l)^2+(H-h)^2} + 13W + 2L + 2H + 2h + 2l \end{aligned} To simplify the formulas by eliminating square roots, we introduce an extra variable s=(L−l)2+(H−h)2s=\sqrt{(L-l)^2+(H-h)^2} which is the slant length of the lid. Then the cost simplifies to: C=40WL+20lW+20hW+20HW+40LH−20(L−l)(H−h)+20Ws+13W+2L+2H+2h+2l\begin{aligned} C&=40WL + 20lW + 20hW + 20HW + 40LH - 20(L-l)(H-h) \\ &\quad+ 20Ws + 13W + 2L + 2H + 2h + 2l \end{aligned} The total volume of the dumpster is the first constraint: V=WLH−W(L−l)(H−h)2=150 V = WLH - \dfrac{W(L-l)(H-h)}{2} = 150 The area of the lid is the second constraint: A=Ws=15 A = Ws = 15 And to account for the extra variable ss, we introduce an extra third constraint: G=s2−(L−l)2−(H−h)2=0 G=s^2-(L-l)^2-(H-h)^2=0 We take the order of the variables in the functions as (W,L,H,l,h,s)(W, L, H, l, h, s). So the gradients are: ∇⃗C=⟨40L+20l+20h+20H+20s+13,40W+40H−20(H−h)+2,20W+40L−20(L−l)+2,20W+20(H−h)+2,20W+20(L−l)+2,20W⟩\begin{aligned} \vec\nabla C &=\big\langle 40L + 20l + 20h + 20H + 20s + 13, \\ &\qquad40W + 40H - 20(H-h) + 2, \\ &\qquad20W + 40L - 20(L-l) + 2, \\ &\qquad20W + 20(H-h) + 2, \\ &\qquad20W + 20(L-l) + 2, \\ &\qquad20W \rangle \\ \end{aligned} ∇⃗V=⟨LH−(L−l)(H−h)2,WH−W(H−h)2,WL−W(L−l)2,W(H−h)2,W(L−l)2,0⟩\begin{aligned} \vec\nabla V &=\left\langle LH - \dfrac{(L-l)(H-h)}{2}, WH - \dfrac{W(H-h)}{2}, \right. \\ &\qquad \left.WL - \dfrac{W(L-l)}{2}, \dfrac{W(H-h)}{2}, \dfrac{W(L-l)}{2}, 0 \right\rangle \\ \end{aligned} ∇⃗A=⟨s,0,0,0,0,W⟩\begin{aligned} \vec\nabla A &=\big\langle s, 0, 0, 0, 0, W \big\rangle \end{aligned} ∇⃗G=⟨0,−2(L−l),−2(H−h),2(L−l),2(H−h),2s⟩\begin{aligned} \vec\nabla G &=\big\langle 0, -2(L-l), -2(H-h), 2(L-l), 2(H-h), 2s \big\rangle \end{aligned} The Lagrange equations, ∇⃗C=λ∇⃗V+μ∇⃗A+κ∇⃗G\vec\nabla C=\lambda\vec\nabla V+\mu\vec\nabla A+\kappa\vec\nabla G, are 40L+20l+20h+20H+20s+13=λ(LH−(L−l)(H−h)2)+μ(s)+κ(0)(1)40W+40H−20(H−h)+2=λ(WH−W(H−h)2)+μ(0)+κ(−2(L−l))(2)20W+40L−20(L−l)+2=λ(WL−W(L−l)2)+μ(0)+κ(−2(H−h))(3)20W+20(H−h)+2=λW(H−h)2+μ(0)+κ(2(L−l))(4)20W+20(L−l)+2=λW(L−l)2+μ(0)+κ(2(H−h))(5)20W=λ(0)+μ(W)+κ(2s)(6)\begin{aligned} 40L + 20l + 20h + 20H + 20s + 13 &=\lambda (LH - \dfrac{(L-l)(H-h)}{2}) + \mu(s) + \kappa (0) \qquad &\text{(1)} \\ 40W + 40H - 20(H-h) + 2 &=\lambda (WH - \dfrac{W(H-h)}{2}) + \mu (0) + \kappa (-2(L-l)) \qquad &\text{(2)} \\ 20W + 40L - 20(L-l) + 2 &=\lambda (WL - \dfrac{W(L-l)}{2}) + \mu (0) + \kappa (-2(H-h)) \qquad &\text{(3)} \\ 20W + 20(H-h) + 2 &=\lambda\dfrac{W(H-h)}{2} + \mu (0) + \kappa (2(L-l)) \qquad &\text{(4)} \\ 20W + 20(L-l) + 2 &=\lambda\dfrac{W(L-l)}{2} + \mu (0) + \kappa (2(H-h)) \qquad &\text{(5)} \\ 20W &=\lambda (0) + \mu (W) + \kappa (2s) \qquad &\text{(6)} \end{aligned} From here, we use a CAS system to solve the system of 99 equations (66 from Lagrange and 33 constraints) in 99 unknowns (W,L,H,l,h,s,λ,μ,κW,L,H,l,h,s,\lambda,\mu,\kappa). The following python code is used to find a numerical solution:

            
    
        def equations(p):
            W, L, H, l, h, s, lam, mu, kap = p
            eq1 = 40*L + 20*l + 20*h + 20*H + 20*s + 13 - lam*(L*H-(L-l)*(H-h)/2) - mu*s - kap*0
            eq2 = 40*W + 40*H - 20*(H-h) + 2 - lam*(W*H - W*(H-h)/2) - mu*(0) - kap*(-2*(L-l))
            eq3 = 20*W + 40*L - 20*(L-l) + 2 - lam*(W*L - W*(L-l)/2) - mu*(0) - kap*(-2*(H-h))
            eq4 = 20*W + 20*(H-h) + 2 - lam*(W*(H-h)/2) - mu*(0) - kap*(2*(L-l))
            eq5 = 20*W + 20*(L-l) + 2 - lam*(W*(L-l)/2) - mu*(0) - kap*(2*(H-h))
            eq6 = 20*W - lam*(0) - mu*(W) - kap*(2*s)
            eq7 = W*L*H - W*(L-l)*(H-h)/2 - 150
            eq8 = W*s - 15
            eq9 = s**2 - (L-l)**2 - (H-h)**2
            return [eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9]
    
        print(fsolve(equations, (4, 4, 4, 1, 1, 1, 1, 1, 1)))
    
          
        

    We use the initial guess (4,4,4,1,1,1,1,1,1)(4, 4, 4, 1, 1, 1, 1, 1, 1) because it is a plausible set of values for our optimal dumpster. It is important to note that your numerical solution may differ depending on your initial conditions. The output of the code produces the values:

          
    
            [4.65, 4.84, 7.20, 2.56, 4.92, 3.22, 17.04, 4.72, 11.03]
    
          
        

    From this output we conclude (W,L,H,l,h,s)=(4.65,4.84,7.20,2.56,4.92,3.22) (W, L, H, l, h, s)=(4.65, 4.84, 7.20, 2.56, 4.92, 3.22) as well as Îť=17.04\lambda=17.04, Îź=4.72\mu=4.72, and Îş=11.03\kappa=11.03. These are plausible dimensions for the dumpster, and thus one optimal solution.

    wll,tj 

    [×]
  48. PY: Checked to here except 20 plot.

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