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