7. Max-Min Problems
Exercises
-
Solution
We compute the partial derivatives, set them equal to and solve for and : So the critical point is . Since and the critical point is a local minimum.
[Ă]Remark
This should have been obvious, since the function is an elliptic paraboloid opening upward with vertex at .
[Ă] -
-
Solution
There are four cases. Three lead to the critical point . The fourth is a contradiction unless again. We apply the Second Derivative test: So Since , the test fails.
[Ă] -
Solution
Equation (2) leads to cases:
- Case 1: : Then equation (1) says and gives the critical point .
- Case 2: : Then equation (1) says or which leads to cases:
- Case 2a: which repeats the critical point .
- Case 2b: and hence and the critical point
The second derivatives are: We make a table to keep track of the values.
Classification test fails minimum -
Solution
To find the critical points, we set the partial derivatives equal to and solve: Equation (1) minus times equation (2) gives . So . Then equation (2) says . This gives the critical point . The second derivatives are: Since and the critical point is a local minimum.
[Ă] -
Solution
To find the critical points, we set the partial derivatives equal to and solve: Equation (1) factors as . This gives us two cases:
- Case 1: : Then equation (2) reduces to or . This gives the critical point .
- Case 2: : Then equation (2) reduces to which factors as . This gives two cases: and . These give the the critical points and .
The second derivatives are: We make a table to keep track of the values.
Classification test fails saddle point -
Solution
To find the critical points, we set the partial derivatives equal to and solve: First notice that equation (2) minus equation (3) gives or . Then equation (2) or (3) reduces to: Next notice that equation (1) factors as . This gives cases:
- Case 1: : Then equation (4) reduces to or . This gives the critical point .
- Case 2: : Using , this says . So equation (4) becomes which expands to which has the solutions and . This gives two critical points and .
To apply the Second Derivative Test, we need the Hessian: We evaluate at each critical point:
- At , The Leading Principal Minor Determinants are They are not all positive and they do not alternate signs and . So this is a saddle point.
- At , The Leading Principal Minor Determinants are So this is also a saddle point.
- At , The Leading Principal Minor Determinants are Again this is a saddle point.
-
Solution
To find the critical points, we set the partial derivatives equal to and solve: Solving equation (1) for gives the equation . Plugging this into equation (2) gives: So the solutions are and .
The second derivatives are: We make a table of values:
Classification Saddle Point Local Maximum wllâ
[Ă] -
Solution
To find the critical points, we set the partial derivatives equal to and solve: Subtracting equation (2) from equation (1) results in the equation which factors as . So the solutions are and .
- Case 1: : Plugging this into equation (1) yields and the critical point .
- Case 2: : Plugging this into equation (1) yields and the critical point .
The second derivatives are: We make a table:
Classification Local Minimum Saddle Point wllâ
[Ă] -
Solution
To find the critical points, we set the partial derivatives equal to and solve: Solving for in equation (1) gives . Plugging this into equation (2) gives: So the solutions are and .
The second derivatives are: We make a table of values:Classification Saddle Point Local Minimum wllâ
[Ă] -
Answer
The critical point, , is a saddle point.
[Ă]
The critical point, , is a local maximum.
The critical point, , is a local minimum.
The critical point, , is a saddle point.Solution
To find the critical points, we set the partial derivatives equal to and solve: Since each first partial derivatve is a function of only one variable, We can solve them separately.
Equation (1) factors as . The solutions are .
Equation (2) factors as . The solutions are .
So there are critical points:The second derivatives are: We make a table:
Classification Saddle Point Local Maximum Local Minimum Saddle Point wllâ
[Ă] -
Answer
The critical point, , is a saddle point.
[Ă]
The critical point, , is a local minimum.
The critical point, , is a local minimum.Solution
To find the critical points, we set the partial derivatives equal to and solve: Solving equation (1) for yields . Plugging this into equation (2) gives: Solutions are and . So the critical points are:
The second derivatives are: We make a table of values:
Classification Saddle Point Local Minimum Local Minimum wllâ
[Ă] -
Answer
The critical point, , is a saddle point.
[Ă]
The critical point, , is a local minimum.
The critical point, , is a local minimum.Solution
To find the critical points, we set the partial derivatives equal to and solve: Solving equation (1) for yields . Plugging this into equation (2) gives: Solutions are and . So the critical points are:
The second derivatives are: We make a table of values:
Classification Saddle Point Local Minimum Local Minimum wllâ
[Ă] -
Find the dimensions and volume of the largest box with base on the plane and upper vertices on the paraboloid .
Solve by eliminating .Solution
The volume of the box is given by where , and are the coordinates of the vertex of the box in the first quadrant. We can eliminate a variable by substituting the constraint equation : Next we set the and partial derivatives equal to zero and solve for and : If or , then the volume would be which would not be the maximum. So taking and , we have We compute times equation (1) minus equation (2): Similarly, times equation (2) minus equation (1) gives: (We ignored the because we want in the first quadrant.) We plug into the parabola: So the length, width, height and volume are:
py,tjâ
[Ă] -
Find the point on the plane that is closest to the origin. Then find the distance from the point to the origin.
Solve by eliminating .
Rotate the second plot with your mouse.1Hint
It is easier to minimize the square of the distance than the actual distance. The answer will be the same.
[Ă]Solution
We need to minimize the distance from the origin to the point on the plane It is easier to minimize the square of the distance, which will give the same solution. So we will minimize the function Since the point must lie on the plane, we substitute the equation of the plane into : We take the and derivatives and set them equal to zero: Equivalently: Multiply the first equation by and the second equation by and add: Substituting back we have: So the closest point is . Its distance from the origin is
py,tjâ
[Ă] -
Find the dimensions and volume of the largest box with base on the plane and upper vertices on the paraboloid .
Solve by using a Lagrange multiplier.Hint
What are the length, width and height in terms of the point on the paraboloid in the first octant.
[Ă]Solution
We need to maximize the volume. If we take the point in the first octant, the height is , but the length and width need to be doubled: and . So the volume is . This is subject to the constraint that the point lies on the paraboloid, . The gradients are: So the Lagrange equations, , are: We multiply (1) by , (2) by , (3) by and equate the left hand sides: This says and . We plug these into the constraint: So So the length, width, height and volume are:
[Ă] -
Find the radius of the sphere with its center at the origin which is tangent to the plane .
Solve by using a Lagrange multiplier.
Rotate the second plot with your mouse.1Hint
The radius of the sphere is the shortest distance to the plane. It is easier to minimize the square of the distance.
[Ă]Solution
The radius of the sphere is the shortest distance to the plane. We minimize the square of the distance: subject to the constraint that the point is on the plane: The gradients are The Lagrange equations, , are Consequently, . Plug these into the constraint: So , and the point of tangency is . So the radius is:
tjâ
[Ă] -
Find the point(s) on the hyperbolic cylinder closest to the origin.
Solution
We minimize the square of the distance function from the origin using on the hyperbolic cylinder . The gradients of the functions are The Lagrange equations, , are Equation (1) yields . Factoring equation (2) gives: The two solution are and .
Case 1: Plugging into the constraint equation , gives: and equation (3) gives ..
Case 2: Plugging into equation (3), gives which says . Then the constraint gives which only has imaginary solutions.
So the two critical points are: which are at a distance from the origin. Any other point would have and so a bigger distance.wllâ
[Ă] -
Find the largest and smallest values of the function on the ellipse .
Answer
The smallest value of along the ellipse is .
[Ă]
The largest value of along the ellipse is .Solution
We maximize and minimize the function with the constraint of the ellipse .
The gradients of the functions are The Lagrange equations, , are Solving for in each equation yields Plugging this result into the constraint equation gives: Plugging these values of back into (*) gives . This results in 4 critical points along the elliptical constraint: We compute the values of : Consequently, the smallest value of is and largest values is .wllâ
[Ă] -
Find the largest and smallest values of the function on the ellipsoid .
In the plot, the ellipsoid is red, the level sets of where are blue, and the level sets of where are yellow.
1Solution
We maximize and minimize the function with the constraint ellipsoid .
The gradients of the functions are The Lagrange equations, , are We multiply the first equation by , the second by and the third by and equate them: These tell us: Plugging these equations into the constraint equation gives: Plugging these values of into (*) gives: Plugging positive and negative variations of into give . So the a minimum value of is and the maximum value is .wll,tjâ
[Ă]
PY: x3d plot not showing.
-
A wire is cut into pieces of lengths and . The piece of length is bent into a square. The piece of length is bent into a rectangle whose length is twice its width. What are and which minimize or maximize the total area enclosed by both pieces? (Allow for the case that or is zero.)
Hint
Express the total area as a function of its segment lengths and . Find the critical points where and are both non-zero and where one is zero. Then make a table of values.
[Ă]Solution
We represent the length of the original wire as a sum of its parts: 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 is folded into a square of side : And now the segment of length is folded into a rectangle of width and length : The total area is then the sum of two areas. We apply the constraint (1) by plugging into our total area function: We set the derivative equal to and solve of : Plugging back this gives: We test the critical points and the endpoint cases (where or are zero) to see which is a minimum or maximum. We plug and into (*) and make a table of values:
Area Classification Local Minimum Neither Local Maximum wll,tjâ
[Ă]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.
[Ă] -
A wire is cut into pieces of lengths , and . The piece of length is bent into a square. The piece of length is bent into a rectangle whose length is twice its width. The piece of length is bent into a rectangle whose length is times its width. What are , and which minimize and maximize the total area enclosed by all pieces? (Allow for cases where , , or are zero.)
Hint
Express the total area as a function of its segment lengths , and . Find the critical points where , and are all non-zero, where one is zero and where two are zero. Then make a table of values.
[Ă]Solution
We express the length of the wire as the sum of its parts: 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 is folded into a square of side : The piece of length is folded into a rectangle of width and length : The piece of length is folded into a rectangle of width and length : The total area is then found by adding the three areas: The gradients of the functions are The Lagrange equations, , are Solving for and in terms of gives: Plugging these results into the constraint gives: And using this value in previous equations yields:So a critical point is .
We also need to check the boundaries. The plane, , intersects the first quadrant in a triangle. So we check the edges (where one of , or is zero and we need to extremize relative to the other variables) and the vertices (where two are zero and other one is ). For the edges, we compute:
First, when :
The Lagrange equations and the constraint lead to: Then when Then when We make a table of values:
Area Classification Local Minimum Neither Neither Neither Neither Neither Local Maximum wll,tjâ
[Ă]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.
[Ă] -
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 meters?
Answer
The dimensions that maximize the area are:
[Ă]
and
where is the length as the partition. These dimensions produce a total area of .Solution
The total fence length is written as a function of width, , and the height including the partition. So the perimeter constraint is: The total enclosed area is: The gradients of the functions are The Lagrange equations, , are Dividing equation (1) by equation (2) yields .
Plugging this into the constrained equation gives: We plug this the formula for to get . These two dimensions produce a total area of .This area is the maximum because is constrained to and the area is at both endpoints while the only interior critical point gives a positive area.
wll,tjâ
[Ă] -
A box manufacturer wants to make a rectangular solid cardboard box with a vertical cardboard divider which splits the interior into compartments, each containing . Because of folding, the top and bottom will have layers of cardboard. What are the dimensions of the box which uses the least amount of cardboard?
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.
[Ă]Solution
Let and be the length and width of the box and let be the height. There are layers of cardboard on the top and bottom with area . There are layers of cardboard in the front and back with area . Finally, there are layers of cardboard on the left, the right and the partition with area . All together, the area is: And the total enclosed volume is: The gradients of the functions are The Lagrange equations, , are We multiply equation (1) by , equation (2) by , and equation (3) by . We then equate and solve: Substituting these values of and into the volume constraint gives: So , and then and .
wllâ
[Ă] -
Find the maximum value of on the line .
Hint
To see the critical point produces a maximum, look at the values of at the nearby point and .
[Ă]Solution
We maximize with the constraint set . The gradients of the functions are
The Lagrange equations, , are Plugging this into the constraint equation gives: So the critical point is where . To see this is a maximum, we look at a near by point with and . Then: So the maximum point of on the line is .
wll,tjâ
[Ă] -
Find the minimum value of on the branch of the hyperbola in the first quadrant. Discuss the geometrical relation between this problem and the previous problem where the constraint and the extremal are interchanged.
Hint
To check the critical point is a minimum, look at the values of at the largest and smallest value of that are allowed.
[Ă]Solution
We minimize with the constraint set . The gradients of the functions are
The Lagrange equations, , are Plugging this into the constraint equation gives: Since we want the point in the first quadrant, the critical points is . To see this is a minimum, we notice that when is very large, then is close to and is very large. Similarly, when is close to , then is very large and is also very large. So there must be a minimum in between and is the only critical point. So, the minimum point of on the hyperbola is .
wll,tjâ
[Ă]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 of this problem is a specific hyperbola. The level sets of the extremal function are the family of all lines with slope . The solution of this problem is the point where one of these lines is tangent to the hyperbola .
Similarly, in the previous problem, the constraint is a specific line. The level sets of the extremal function are the family of all hyperbolas of the form . The solution of that problem is the point where one of these hyperbolas is tangnet to the line .
The differ because the constraint and the extremal are interchanged.
[Ă] -
Find the maximum volume of an cylindrical aluminum can with lids whose surface area is .
Hint
Construct functions for the volume and surface area of the can. Remember the area of the two ends. Treat and as you would and .
[Ă]Solution
The curved part of the can has area while the top and bottom have area . So we want the total area to be . We maximize the volume function, subject to the area constraint. The gradients of the functions are The Lagrange equations, , are Dividing equation (1) by equation (2) gives: Substituting this into the area constraint gives Therefore the critical point , produces a maximized volume of .
wll,tjâ
[Ă] -
Find the minimum surface area of a cylindrical aluminum can with lids whose volume is . Discuss the geometrical relation between this problem and the previous problem.
Hint
Construct functions for the volume and surface area of the can. Remember the area of the two ends. Treat and as you would and .
[Ă]Solution
We minimize the surface area function, , with the constraint of a fixed volume, .
The gradients of the functions are The Lagrange equations, , are Dividing equation (1) by equation (2) gives: Substituting this into the volume constraint gives Therefore the critical point , produces a minimized surface area .wll,tjâ
[Ă]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 . The level sets of the extremal function 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 . The level sets of the extremal function 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.
[Ă] -
Find the point(s) on the curve closest to the origin and give the distance from the origin.
Solution
We minimize the square of the distance function from the origin using on the constraint curve . The gradients of the functions are The Lagrange equations, , are We divide (1) by (2) giving or . Plugging this into the constraint yields: So the two minima points are: which are at a distance from the origin.
wll,tjâ
[Ă] -
Find the point on the surface closest to the origin.
Solution
We minimize the square of the distance function from the origin using on the constraint curve . The gradients of the functions are The Lagrange equations, , are We divide (1) by (3) giving or . We also divide (2) by (3) giving or . Plugging this into the constraint yields: So the the minima points are: which are at a distance from the origin.
wll,tjâ
[Ă] -
Find the volume of the largest rectangular solid with faces in the coordinate planes and the opposite vertex on the plane where , and .
Hint
Write the volume of the solid described in the question as a function of , , and Then you can use Lagrange equations to solve for the maximized volume.
[Ă]Solution
We maximize the volume, where is the vertex on the plane . The gradients of the functions are The Lagrange equations, , are We equate lambda in each equation, which yields the following equations: Substituting these values into the equation of the plane produces: We use this value to solve for and . So the critical point is: With a corresponding maximized volume of
wll,tjâ
[Ă] -
We would like to find the line which best fits a collection of data points using a quadratic fit. For each point, , the vertical distance to the line is . Click to toggle the vertical line segments whose lengths are . To say we want a quadratic fit, means we want to find and so that the sum of the squared vertical distances is a minimum. Find formulas for and in terms of the 's and 's. What happens if we have more than points?
Hint
Note: The 's and 's are constants. The variables are and . It may help to start with fewer points, say , so you can write out all the terms in the summation.
[Ă]Solution
We treat as a function, , where and are constants. We substitute : We take the partial derivatives with respect to : and with respect to : To simplify the computation, we introduce the notation: To minimize we set (half) the partial derivatives equal to : And we solve this system of linear equations for and . Solving equation (2) for gives: Plugging this value of into equation (1) gives: We solve for : And plugging this value of back into the solution for gives: This produces the final solutions:
General formulas are found by repeating the process with rather than . where
wll,tjâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function on the circle .
In the plot, the circle is red and the level sets of are blue.
-
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.
[Ă]Solution
We check for critical points on the boundary circle using the Lagrange method. The gradients of the functions are The Lagrange equations, , are We divide equation (1) by equation (2) to get which implies . We plug this into the contraint We use this value to solve for : (Note they must have the same sign.) So our critical points are: We make a table of values:
So the global maximum is at and the global minimum is at .
wll, tjâ
[Ă]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 and
the minimum . -
Solve by parametrizing the constraint.
Solution
We parametrize the circle as . Then the extremal function becomes: We set the derivative equal to and solve for : Since the tangent is positive, the angle must be in the or quadrants. So the solutions are: Finally, we evaluate the extremal at the points: So the global maximum is and the global minimum is .
tjâ
[Ă]
-
-
An aquarium is to have a volume of . Its base will be made of marble which costs $ per . The front will be glass which costs $ per . The sides and back will be aluminum which costs $ per . What are the dimensions of the cheapest such aquarium?
Hint
Construct two functions, the cost of the aquarium (optimizing), and the volume of the aquarium (constraint) as functions of the dimensions.
[Ă]Solution
We define ,, and such that the area of the base is , the front is , and the sides are . We then create a function for the cost of the aquarium: With the defined contraint volume of , we minimize with contraint using the Lagrange method. The gradients of the functions are The Lagrange equations, , are We multiply equation (1) by , equation (2) by , and equation (3) by . We can then equate the left side of each equation: We solve the first and second equations: Plugging these solutions for and into gives: Using this solution of yields and . Therefore the cheapest dimensions of the aquarium are
wll,tjâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside or on the triangle with vertices , and .
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.
[Ă]Solution
To find the critical points inside the triangle, we set the partial derivatives of equal to and solve: From the plot, we see the critical point lies within the traingle.
Next, we find all the critical points along the edges of the triangle region, represented by , , and . We set the partial derivatives equal to zero with each respective constraint. For : This implies that is a critical point on . Next, for : This implies that is a critical point on . Next, for : This implies that is a critical point on .
We make a table of values of for each critical point inside, on the edges and at each vertex.
â â So from the table we can conclude that the minimum of the function in the traingular region is at , and the maximum is at
wllâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside or on the rectangle with vertices , , and .
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.
[Ă]Solution
To find the critical points inside the rectangle, we set the partial derivatives of equal to and solve: The critical point lies within the rectangle and .
Next, we find all the critical points along the edges of the rectangle, which are , , , and . We restrict the function to each edge and then set the derivative equal to zero and solve.
For : For : For : For : We make a table of values for each critical point we found, along with each vertex.Region Classification Interior Minimum Edge Edge Edge Edge Vertex Maximum Vertex Vertex Vertex So from the table we can conclude that the minimum of the function is at the interior point , and the maximum is at the vertex .
wllâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside or on the square and .
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 and are within the region.
[Ă]Solution
To find the critical points inside the square, we set the partial derivatives of equal to and solve. Notice that on , and on . The critical point lies within the square.
Next, we find all the critical points along the edges of the square, represented by , , , and . We evaluate the function on each edge and then equate the derivative to zero and solve.
For : For : For : For : We make a table of values for each critical point we found, along with each vertex.Region Classification Interior Maximum Edge Edge Edge Edge Vertex Minimum Vertex Minimum Vertex Minimum Vertex Minimum So from the table we can conclude that the minimum of the function is at the verticies, and the maximum is at in the interior.
wllâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside or on the quarter of the circle in the 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.
[Ă]Answer
The minimum of the function in the quarter circle is at , and the maximum is at along the arc.
[Ă]Solution
To find the critical points inside the quarter circle, we set the partial derivatives of equal to and solve. 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 or . We evaluate the function on each edge and set the derivatve equal to zero.
For : For : Again this implies that there is no critical point on orFinally, we look for critical points along the arc of the circle. This will be the point where the level set of is tangent to the circle, as shown in the plot. This is best found using Lagrange multipliers. The gradients of the functions are: The Lagrange equations, , are
We divide equation (1) by equation (2) to get which implies . We plug this into the contraint to get: We use the positive solution, , to solve for because our arc is in the first quadrant. So our only critical point is . We make a table of values of for each critical point and each vertex:
Region Classification Edge Maximum Vertex Minimum Vertex Vertex So from the table we can conclude that the minimum of the function in the quarter circle is at , and the maximum is at along the arc.
wllâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside or on the eighth of the sphere in the octant.
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.
[Ă]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 equal to and solve. 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 , , and . We evaluate the function on each surface and set the derivatve equal to zero. This implies that there is no critical point on any of the flat surfaces.
Next, we can use Lagrange multipliers with the contraint surface to find critical points along the spherical surface of the object. The gradients of the functions are: The Lagrange equations, , are We divide equations (1) and (2) by equation (3) and solve in terms of . We plug these into the contraint We use the positive solution, , to solve for and because our object is in the first octant. So the critical point on the spherical surface is .
Next, we must find the critical points along the edges of the object. The straight edges are represented by , , and . We evaluate the function at these lines and set the derivatives equal to zero. 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 , on . The Lagrange equations say: So the constraint says: For , on . The Lagrange equations say: So the constraint says: For , on . The Lagrange equations say: So the constraint says: We make a table of values of for each critical point we found, along with each vertex.Region Classification Surface Maximum Edge Edge Edge Vertex Vertex Vertex Vertex Minimum So from the table, we can conclude that the minimum of the function is at the vertex , and the maximum is at the point on the spherical surface. In the plot, the level sets of are the planes in blue or orange. The upper orange level set is for which is tangent to the sphere at . The lower orange level set is for which passes through the point . All the points in the interior have values of between and .
wll,tjâ
[Ă] -
Consider the function and the square and .
In the plot, the square is red and the level sets of are blue.
-
Find the locations and values of the global maximum and minimum of the function on the (edges of the) square.
Hint
Find the critical points on each edges. Then make a table of the values of at those points and the vertices to determine the minimum and maximum locations. When solving for critical points, be sure and are within the region.
[Ă]Solution
We find the critical points along all the edges of the square, represented by , , , and . We evaluate the function on each edge and then equate the derivative to zero and solve.
For : This is not on the edge of the square because does not satisfy .
For : For : For : This is not on the edge of the square because does not satisfy .We make a table of values of for each critical point we found, along with each vertex:
Region Classification Edge Edge Minimum Vertex Vertex Maximum Vertex Vertex So from the table we conclude that the minimum of the function in the square is at the edge point , and the maximum is at the vertex .
In the animation, the square is red,
and the level sets of are blue.
The level sets for the minimum and maximum on the square flash in orange.wll,tjâ
[Ă] -
Find the locations and values of the global maximum and minimum of the function inside and on the square.
Hint
Find the critical points in the interior and the edges. Then make a table of the values of at those points and the vertices to determine the minimum and maximum locations. When solving for critical points, be sure and are within the region.
[Ă]Solution
To find the critical points inside the square, we set the partial derivatives of equal to and solve. We add the two equations to eliminate terms, producing the equation . This implies that our critical point is which we observe is inside the square.
We add this to the table of values we found in part (a):
Region Classification Interior Minimum Edge Edge Vertex Vertex Maximum Vertex Vertex So from the table we can conclude that the minimum of the function in or on the square is at the interior point , and the maximum is at the vertex .
In the animation, the square is red,
and the level sets of 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.wll,tjâ
[Ă]
-
-
The plane intersects the cylinder in an ellipse. Find the points on this ellipse closest and farthest from the origin.
-
Solve using Lagrange multipliers.
-
Solve using a paramaterization.
1Hint
Simplify your calculations by optimizing the squared distance. Find the gradients of the distance squared and the two contraints. Then, solve the Lagrange equations, , along with the constraints to find critical points. Finally, make a table of values to determine the distances of the critical points found.
[Ă]Solution
We minimize the square of the distance function from the origin using on the intersection of the plane and the cylinder The gradients of the functions are The Lagrange equations, , are Equation (3) implies and we eliminate by plugging it into equations (1) and (2). To eliminate , we multiply equation (4) by and equation (5) by and subtract. This yields: There are two scenarios, where , or . We use both constraints to find critical points.
Case 1: The last equation says or . Then or respectively. So the two critical points are: Case 2: or equivalently So the two critical points are: We make a table of values to determine the distance of the critical points.Classification Minimum Minimum Maximum So from the table, we conclude that the points closest to the origin is and . The point farthest from the origin is .
wll,tjâ
[Ă]Hint
Simplify your calculations by optimizing the squared distance. Start by paramaterizing the cylinder in the - plane.
[Ă]Solution
Using cyclindrical coordinates, we can paramaterize on the ellipse. Since the raidus of the cylinder is : We use the plane equation to solve for and paramaterize: Substituting these values into : To find the critical points We differentiate and equate it to zero: Case 1: Case 2: We make a table of values of to determine the distance of the critical points.
Classification Minimum Minimum Maximum So from the table, we conclude that the points closest to the origin is and . The point farthest from the origin is .
wll,tjâ
[Ă] -
-
Maximize the function on the line of intersection of the planes and .
Hint
Determine the geometry of the constraints and attempt to parameterize the intersection. Maximize the function with respect to the parameterized variable.
[Ă]Solution
To maximize the function , we parameterize the function along the line of intersection of the planes and , as follows: We plug the parameterized variables into : To find the critical points along this intersection, we take the derivatve of with respect to and equate it to zero: To determine if this is a minimum or maximum, we use the Second Derivative Test. Since is always less than zero, the critical point at is a maximum. We use to get cartesian coordinates and the maximum value of . So the maximum value of at the intersection of the planes and is .
wllâ
[Ă] -
A box manufacturer wants to make a rectangular solid cardboard box to contain . The box is folded from a single rectangular piece of cardboard so that the flaps from sides just meet at the centerline of the top and bottom. The flaps from the other 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?
Hint
Let and be the length and width of the original piece of cardboard. Let , and 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?
[Ă]Answer
The dimensions of the original carboard piece that minimize the amount of cardboard used are: and .
[Ă]Solution
Let , and be the length, width and height of the final box, resp. Let and be the length and width of the original piece of cardboard, resp. The length of the orignal cardboard is the perimeter of the top, so it can be written as . 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 . Therefore, the amount of cardboard used is: And the total enclosed volume is: The gradients of the functions are The Lagrange equations, , are We multiply equation (1) by , equation (2) by , and equation (3) by . We then equate and solve: Substituting into the volume constraint gives: We substitute this into and solve for : This is a quadratic equation for . We use the quadratic formula: We take the positive solution . So , and . So the dimensions of the original carboard piece that minimixes the amount of cardboard used are:
wllâ
[Ă] -
A paper cup has the shape of a frustum of a cone with a bottom. Let the smaller bottom radius be , the larger top radius be and the height be . If the cup needs to hold of water, what are the dimensions , and which use the least paper?
Hint
The area of the slanted sides of the cup (frustum) is: The area of the bottom is . So the total area is: The volume of the cup (frustum) is Add an extra constraint which says the slant height is so then the total area is: This will eliminate a lot of square roots from the problem.
[Ă]Solution
From the hint, the surface area of the cup is where the slant height satisfies . So we want to minimize the area subject to the constraint that the volume is constant: and the constraint on the slant height: We take the order of the variables in the functions as . So the gradients are: The Lagrange equations, , are 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 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 as well as and . These are plausible dimensions for the frustum, and thus one optimal solution.
wll,tjâ
[Ă] -
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 be the width across the front. Let be the length of the base from back to front. Let be the height of the back. Let be the length of the top from the back to the lid. Let be the height of the front from the bottom to the lid. The heavy steel base costs $ per square foot. The sides, top and lid are lighter steel which cost $ per square foot. Every edge needs to be welded which costs $ per foot. The hinge for the lid costs $ per foot including its welding cost. The volume needs to be . The lid needs to be . What are the dimensions which minimize the total cost? (Solving the resulting equations may require a computer algebra system.)
Hint
To eliminate a bunch of square roots, it is helpful to introduce an extra variable which is the slant length of the lid: along with the extra constraint: Find the set of Lagrange equations then use a CAS system to find a numerical solution.
[Ă]Solution
We construct a function for the total cost of construction. First, at $ per square foot, the base costs: At $ per square foot, the top, front, back, sides, and lid cost: At $ per foot, the hinge costs: At $ per foot, the welding costs: So the total cost of the dumpster is: To simplify the formulas by eliminating square roots, we introduce an extra variable which is the slant length of the lid. Then the cost simplifies to: The total volume of the dumpster is the first constraint: The area of the lid is the second constraint: And to account for the extra variable , we introduce an extra third constraint: We take the order of the variables in the functions as . So the gradients are: The Lagrange equations, , are From here, we use a CAS system to solve the system of equations ( from Lagrange and constraints) in unknowns (). 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 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 as well as , , and . These are plausible dimensions for the dumpster, and thus one optimal solution.
wll,tjâ
[Ă]
PY: Checked to here except 20 plot.
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.
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.
Heading
Placeholder text: Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum