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# 12.4 The Mean Value Theorem

12.24   Lemma (Rolle's Theorem) Let be real numbers with and let be a function that is continuous on and differentiable on . Suppose that . Then there is a point such that .

Proof: By the extreme value property, has a maximum at some point . If , then by the critical point theorem. Suppose . By the extreme value property, has a minimum at some point . If then by the critical point theorem. If , then we have so . Hence in this case the maximum value and the minimum value taken by are equal, so for so for all .

Rolle's theorem is named after Michel Rolle (1652-1719). An English translation of Rolle's original statement and proof of the theorem can be found in [43, pages 253-260]. It takes a considerable effort to see any relation between what Rolle says and what our form of Rolle's theorem says.

12.25   Theorem (Mean value theorem.) Let be real numbers and let be a function that is continuous on and differentiable on . Then there is a point such that ; i.e., there is a point where the slope of the tangent line is equal to the slope of the line joining to .

Proof: The equation of the line joining to is

Let

Then is continuous on and differentiable on and . By Rolle's theorem there is a point where .

Now

so

12.26   Corollary. Let be an interval in and let be a function that is continuous on and differentiable at the interior points of . Then

Proof: I will prove the second assertion. Suppose for all . Let be points in with . Then by the mean value theorem

Since and , we have ; i.e., . Thus is decreasing on

12.27   Exercise. A Prove the first assertion of the previous corollary; i.e., prove that if is continuous on an interval , and for all , then is constant on .

12.28   Definition (Antiderivative) Let be a real valued function with . Let be an interval such that . A function is an antiderivative for on if is continuous on and for all in the interior of .

12.29   Examples. Since , we see that is an antiderivative for . Since

and

we see that and are both antiderivatives for .

We will consider the problem of finding antiderivatives in chapter 17. Now I just want to make the following observation:

12.30   Theorem (Antiderivative theorem.) Let be a real valued function with and let be an interval with . If and are two antiderivatives for on , then there is a number such that

12.31   Exercise. A Prove the antiderivative theorem.

12.32   Definition (Even and odd functions.) A subset of is called symmetric if . A function is said to be even if is a symmetric subset of and

and is said to be odd if is a symmetric subset of and

.

12.33   Example. If and , then is even if is even, and is odd if is odd. Also is an even function and is an odd function, while is neither even or odd.

12.34   Example. If is even, then where is the reflection about the vertical axis. If is odd, then where is a rotation by about the origin.

12.35   Exercise. A Are there any functions that are both even and odd?

12.36   Exercise. A
a)
If is an arbitrary even differentiable function, show that the derivative of is odd.
b)
If is an arbitrary odd differentiable function, show that the derivative of is even.

Next: 13. Applications Up: 12. Extreme Values of Previous: 12.3 Maxima and Minima   Index
Ray Mayer 2007-09-07