Next: 12.4 The Mean Value
Up: 12. Extreme Values of
Previous: 12.2 A Nowhere Differentiable
Index
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
![$f\colon [0,1]\to\mbox{{\bf R}}$](img3164.gif)
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
![$[a,b]$](img1071.gif)
, then

has a maximum and a minimum on
![$[a,b]$](img1071.gif)
.
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.
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
![$[-2,2]$](img3194.gif)
we know that

has a
maximum and a minimum on
![$[-2,2]$](img3194.gif)
. 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.
Next: 12.4 The Mean Value
Up: 12. Extreme Values of
Previous: 12.2 A Nowhere Differentiable
Index
Ray Mayer
2007-09-07