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
![$f$](img676.gif)
is continuous on an interval
![$J$](img1131.gif)
, and
![$f^\prime (x)=0$](img3220.gif)
for all
![$x\in\mbox{interior} (J)$](img3234.gif)
,
then
![$f$](img676.gif)
is constant on
![$J$](img1131.gif)
.
12.28
Definition (Antiderivative)
Let
![$f$](img676.gif)
be a real valued function with
![$\mbox{{\rm dom}}(f)\subset\mbox{{\bf R}}$](img2672.gif)
. Let
![$J$](img1131.gif)
be an
interval such that
![$J\subset\mbox{{\rm dom}}(f)$](img3242.gif)
. A function
![$F$](img162.gif)
is an
antiderivative for ![$f$](img676.gif)
on
![$J$](img1131.gif)
if
![$F$](img162.gif)
is continuous on
![$J$](img1131.gif)
and
![$F^\prime
(x)=f(x)$](img3243.gif)
for all
![$x$](img35.gif)
in
the interior of
![$J$](img1131.gif)
.
12.29
Examples.
Since
![$\displaystyle {{ d \over dx} (x^3 + 4) = 3x^2}$](img3244.gif)
, we see that
![$x^3 + 4$](img3245.gif)
is an
antiderivative for
![$3x^2$](img3246.gif)
.
Since
and
we see that
![$\cos^2$](img3249.gif)
and
![$-\sin^2$](img3250.gif)
are
both antiderivatives for
![$-2\sin\cdot\cos$](img3251.gif)
.
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
![$S$](img49.gif)
of
![$\mbox{{\bf R}}$](img153.gif)
is called
symmetric if
![$(x\in S\mbox{$\hspace{1ex}\Longrightarrow\hspace{1ex}$}-x\in S)$](img3253.gif)
. A
function
![$f$](img676.gif)
is said to be
even if
![$\mbox{{\rm dom}}(f)$](img2675.gif)
is a symmetric subset of
and
and
![$f$](img676.gif)
is said to be
odd if
![$\mbox{{\rm dom}}(f)$](img2675.gif)
is a symmetric subset of
![$\mbox{{\bf R}}$](img153.gif)
and
.
12.33
Example.
If
![$n\in\mbox{${\mbox{{\bf Z}}}^{+}$}$](img282.gif)
and
![$f(x)=x^n$](img3257.gif)
, then
![$f$](img676.gif)
is even if
![$n$](img9.gif)
is even, and
![$f$](img676.gif)
is odd if
![$n$](img9.gif)
is odd. Also
![$\cos$](img2398.gif)
is an even function and
![$\sin$](img2401.gif)
is an odd
function,
while
![$\ln$](img1348.gif)
is neither even or odd.
12.34
Example.
If
![$f$](img676.gif)
is even, then
![$V\Big({\rm graph} (f)\Big)={\rm graph} (f)$](img3258.gif)
where
![$V$](img1205.gif)
is
the reflection about the vertical axis. If
![$f$](img676.gif)
is odd, then
![$R_\pi \Big({\rm graph}(f)\Big)={\rm graph}(f)$](img3259.gif)
where
![$R_\pi$](img1206.gif)
is a rotation by
![$\pi$](img5.gif)
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