16. Max-Min Problems

a. Local Minima, Local Maxima and Saddle Points

1. Functions of \(1\) Variable

We first review the definitions of a local maximum and a local minimum for functions of \(1\) variable.

For a function of \(1\) variable, \(f(x)\), a local or relative maximum (resp. local or relative minimum) occurs at a point, \(x=c\), where the function value is relatively larger (resp. smaller) than those at all . Together, local minima and maxima are called local or relative extrema.

When we say nearby, we actually mean that there is some interval around \(c\) within which \(f(c)\) is the largest (resp. smallest) value. We don't care what happens far away from \(c\).

For a function of \(1\) variable, \(f(x)\), extrema occur when the tangent line is horizontal or the derivative does not exist. We search for extrema by identifying critical points (points, \(c\), at which the derivative, \(f'(c)\), is zero or undefined) and classifying each as a local minimum, a local maximum or a horizontal inflection point (at which nearby values are larger on one side and smaller on the other). There are no saddle points for functions of \(1\) variable.

Assuming all derivatives exist, we distinguish between these three possibilities by first looking at the second derivative:

  1. If \(f''(c) > 0\), then the critical point \(c\) is a local minimum.
  2. If \(f''(c) < 0\), then the critical point \(c\) is a local maximum.
  3. If \(f''(c)=0\), then the TEST FAILS.
The graphic depicts the 2D plot of a concave up parabola with its vertex at x = 2. The parabola opens upward, making the critical point a local minimum.
Local Minimum
\(f''(c) \gt 0\)
The graphic depicts the 2D plot of a concave down parabola with its vertex at x = 2. The parabola opens downward, making the critical point a local maximum.
Local Maximum
\(f''(c) \lt 0\)\

Here are four functions which all have a critical point at \(x=2\) and all satisfy \(f''(2)=0\). However, one has a local minimum, one has a local maximum and two have horizontal inflection points. So the condition \(f''(2)=0\) cannot distinguish between the three cases.

The graphic depicts the 2D plot of a concave up fourth degree polynomial with a local minimum at the critical point x = 2.
\(\begin{aligned} f(x)&=(x-2)^4+2 \\ f'(x)&=4(x-2)^3 \\ f'(2)&=0 \\ f''(x)&=12(x-2)^2 \\ f''(2)&=0 \end{aligned}\)
The graphic depicts the 2D plot of a concave down fourth degree polynomial with a local maximum at the critical point x = 2.
\(\begin{aligned} f(x)&=-(x-2)^4+2 \\ f'(x)&=-4(x-2)^3 \\ f'(2)&=0 \\ f''(x)&=-12(x-2)^2 \\ f''(2)&=0 \end{aligned}\)
The graphic depicts the 2D plot of an increasing cubic polynomial with a horizontal inflection point at the critical point x = 2.
\(\begin{aligned} f(x)&=(x-2)^3+2 \\ f'(x)&=3(x-2)^2 \\ f'(2)&=0 \\ f''(x)&=6(x-2) \\ f''(2)&=0 \end{aligned}\)
The graphic depicts the 2D plot of a decreasing cubic polynomial with a horizontal inflection point at the critical point x = 2.
\(\begin{aligned} f(x)&=-(x-2)^3+2 \\ f'(x)&=-3(x-2)^2 \\ f'(2)&=0 \\ f''(x)&=-6(x-2) \\ f''(2)&=0 \end{aligned}\)

When the Second Derivative Test fails, we turn to the first derivative:

  1. If \(f'(x) < 0\) for \(x < c\) and \(f'(x) > 0\) for \(x > c\), then the critical point \(c\) is a local minimum.
  2. If \(f'(x) > 0\) for \(x < c\) and \(f'(x) < 0\) for \(x > c\), then the critical point \(c\) is a local maximum.
  3. If \(f'(x) < 0\) for \(x < c\) and \(x > c\), or \(f'(x) > 0\) for \(x < c\) and \(x > c\), then the critical point \(c\) is an inflection point.

When we specify the sign of \(f'\) on \(x < c\) and \(x > c\), we actually mean that there is some interval around \(c\) within which these conditions on \(f'\) hold. For local extrema, we don't care what happens far away from \(c\).

The graphic depicts the same plot of a concave up parabola with a local minimum at the critical point x = 2. To the left of x = 2, the slope is negative. To the right of x = 2, the slope is positive.
\(\begin{aligned} f'(x)&\lt0 \,\text{on left} \\ f'(x)&\gt0 \,\text{on right} \\ \end{aligned}\)
Local Minimum
The graphic depicts the same plot of the concave down parabola with a local maximum at the critical point x = 2. To the left of x = 2, the slope is positive. To the right of x = 2, the slope is negative.
\(\begin{aligned} f'(x)&\gt0 \,\text{on left} \\ f'(x)&\lt0 \,\text{on right} \\ \end{aligned}\)
Local Maximum
The graphic depicts the same plot of the increasing cubic polynomial with a horizontal inflection point at the critical point x = 2. On both sides of x = 2, the slope is positive.
\(\begin{aligned} f'(x)&\gt0 \,\text{on left} \\ f'(x)&\gt0 \,\text{on right} \\ \end{aligned}\)
Inflection Point
The graphic depicts the same plot of the decreasing cubic polynomial with a horizontal inflection point at the critical point x = 2. On both sides of x = 2, the slope is negative.
\(\begin{aligned} f'(x)&\lt0 \,\text{on left} \\ f'(x)&\lt0 \,\text{on right} \\ \end{aligned}\)
Inflection Point

We now generalize these ideas to two or more variables.

© MY Math Apps

Supported in part by NSF Grant #1123255

NSF logo