** Next:** 13. Applications
** Up:** 12. Extreme Values of
** Previous:** 12.3 Maxima and Minima
** Index**

** 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:

** 12.31**
**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