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13.1
Example.
Let
. Here
and
is an odd function. We have
From this we see that the critical set for
is
.
We can determine the sign of
by looking at the signs of its
factors:
Since
is odd, I will consider only points where
.
Thus
is strictly increasing on
and on
, and
is
strictly decreasing on
. Also
We see that
is unbounded on any interval
or
,
since the numerator of the fraction is near to
, and the denominator is near
to
on these intervals. Also
so
is large when
is large. (
is the product of
and a
number
near to
.) Using this information we can make a reasonable sketch of the
graph
of
.
Here has a local maximum at and a local minimum at .
It has no global extreme points.
13.2
Definition (Infinite limits.)
Let
be a real sequence. We say
if for every
there is an
such that for all
.
We say
if for every
there is an
such that for all
.
Let
be a real valued function such that
, and let
. We say
if
contains an interval
and for every sequence
in
We say
if
contains an interval
and for every sequence
in
Similar definitions can be made for
We say
if
contains some
interval
and for every sequence
in
Similarly if
we can define
13.3
Example.
If
is the function in the previous example (i.e.
) then
and
Also,
and
The situation here is very similar to the situation in the case of ordinary
limits,
and we will proceed without writing out detailed justifications.
13.4
Exercise.
Write out definitions for
13.5
Exercise.
Find one function
satisfying all of the following conditions:
13.6
Example.
Let
Then
for all
,
so I will restrict my attention to the interval
.
Also
is an odd function, so I will further restrict my attention to the
interval
. Now
Hence
is a critical point for
if and only if
. The critical points of
in
are thus
and
, and the critical points in
are
. Now
and
and
. Also note
. Since
is continuous on
, we know that
has a
maximum and a minimum on this interval, and since
for all
,
the maximum
(or minimum)
of
on
will be a global maximum (or minimum) for
. Since
is
differentiable everywhere, the extreme points are critical points and from our
calculations
has a maximum at
and a minimum at
. I will now determine the sign of
on
:
Thus
is strictly increasing on
and
is
strictly decreasing on
. We can now make a
reasonable sketch for the graph of
.
13.7
Exercise.
Sketch and discuss the graphs of the following functions. Mention all
critical points and determine whether each critical point is a local or global
maximum or minimum.
- a)
-
.
- b)
-
.
- c)
-
.
- d)
- .
(The following remark may be helpful for
determining
. If
, then
. Hence if
, then
Thus,
Next: 13.2 Optimization Problems.
Up: 13. Applications
Previous: 13. Applications
  Index
Ray Mayer
2007-09-07