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# 12.3 Maxima and Minima

12.10   Definition (Maximum, minimum, extreme points.) Let be a set, let and let . We say that has a maximum at if

and we say that has a minimum at if

Points where has a maximum or a minimum are called extreme points of .

12.11   Example. Let be defined by
 (12.12)

Then has a minimum at and at , but has no maximum. To see that has no maximum, observe that if then and

If is the function whose graph is shown, then has a maximum at , and has minimums at and .

12.13   Assumption (Extreme value property.) If is a continuous function on the interval , then has a maximum and a minimum on .

The extreme value property is another assumption that is really a theorem, (although the proof requires yet another assumption, namely completeness of the real numbers.)

The following exercise shows that all of the hypotheses of the extreme value property are necessary.

12.14   Exercise. A
a)
Give an example of a continuous function on such that has no maximum on .
b)
Give an example of a bounded continuous function on the closed interval , such that has no maximum on
c)
Give an example of a function on such that has no maximum on .
d)
Give an example of a continuous function on that has neither a maximum nor a minimum on , or else explain why no such function exists.

12.15   Exercise. A
a)
Show that every continuous function from an interval to is bounded. (Hint: Use the extreme value property,)
b)
Is it true that every continuous function from an open interval to is bounded?
c)
Give an example of a function from to that is not bounded.

12.16   Definition (Critical point, critical set.) Let be a real valued function such that . A point is called a critical point for if . The set of critical points for is the critical set for . The points in the critical set for correspond to points where the graph of has a horizontal tangent.

12.17   Theorem (Critical point theorem I.) Let be a real valued function with . Let . If has a maximum (or a minimum) at , and is differentiable at , then .

Proof: We will consider only the case where has a maximum. Suppose has a maximum at and is differentiable at . Then is an interior point of so we can find sequences and in such that , , for all , and for all .

Since has a maximum at , we have and for all . Hence

Hence by the inequality theorem for limits,

It follows that .

12.18   Definition (Local maximum and minimum.) Let be a real valued function whose domain is a subset of . Let . We say that has a local maximum at if there is a positive number such that

and we say that has a local minimum at if there is a positive number such that

Sometimes we say that has a global maximum at to mean that has a maximum at , when we want to emphasize that we do not mean local maximum. If has a local maximum or a local minimum at we say has a local extreme point at .

12.19   Theorem (Critical point theorem II.) Let be a real valued function with . Let . If has a local maximum or minimum at , and is differentiable at , then .

Proof: The proof is the same as the proof of theorem 12.17.

12.20   Examples. If has a maximum at , then has a local maximum at .

The function whose graph is shown in the figure has local maxima at and local minima at , and . It has a global maximum at , and it has no global minimum.

From the critical point theorem, it follows that to investigate the extreme points of , we should look at critical points, or at points where is not differentiable (including endpoints of domain ).

12.21   Example. Let for . Then is differentiable everywhere on except at and . Hence, any local extreme points are critical points of or are in . Now

From this we see that the critical set for is . Since is a continuous function on a closed interval we know that has a maximum and a minimum on . Now

Hence has global maxima at and , and has global minima at and . The graph of is shown.

12.22   Example. Let

Here and clearly for all . I can see by inspection that has a maximum at ; i.e.,

I also see that , and that is strictly decreasing on

thus has no local extreme points other than . Also is very small when is large. There is no point in calculating the critical points here because all the information about the extreme points is apparent without the calculation.

12.23   Exercise. Find and discuss all of the global and local extreme points for the following functions. Say whether the extreme points are maxima or minima, and whether they are global or local.
a)
for .
b)
for .

Next: 12.4 The Mean Value Up: 12. Extreme Values of Previous: 12.2 A Nowhere Differentiable   Index
Ray Mayer 2007-09-07