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Index
15.24
Definition (Convexity)
Let
be a differentiable function on an interval
. We say that
is
convex upward over or that
holds water over
if and only if for each point
in
, the tangent line to graph(
) at
lies below the graph of
.
Since the equation of the tangent line to graph(
) at
is
the condition for
to be convex upward over
is that for all
and
in

(15.25) 
Condition (
15.25) is equivalent to the two conditions:
and
These last two conditions can be written as the single condition

(15.26) 
We say that is convex downward over , or that spills
water over if and only if for each point in ,
the tangent line to graph() at
lies above the graph of .
This condition is equivalent to the condition that for all points
15.27
Theorem.
Let be a differentiable function over the interval .
Then is convex upward over if and only if is increasing
over . (and similarly is convex downward over if and only
if is decreasing over .)
Proof: If is convex upward over , then it follows from
(15.26) that is increasing over .
Now suppose that is increasing over . Let be distinct points
in . By the mean value theorem there is a point between and
such that
If then so since is increasing over
i.e.
Thus condition (15.26) is satisfied, and is convex upward over
.
15.28
Corollary.
Let be a function such that exists for
all in the interval . If for all
then is convex upward over . If for all
then is convex downward over .
15.29
Exercise.
A
Prove one of the two statements in
corollary
15.28.
15.30
Lemma (Converse of corollary 12.26) Let be a real
function such that is continuous on and differentiable on
. If is increasing on , then for all
.
Proof: let . Choose such that
. Then
is a sequence such that
and hence
Since is increasing on , we have
for all
, and it follows that
15.31
Definition (Inflection point)
Let
be a real function,
and let
. We say that
is a
point of inflection
for
if there is some
such that
, and
is convex upward on one of the intervals
,
, and
is convex downward on the other.
15.32
Theorem (Second derivative test for inflection points) Let be a real function, and let be a point of inflection for . If is defined and continuous in some
interval
then
Proof: We will suppose that is convex upward on the interval
and is convex downward on . (The proof in the
case where these conditions are reversed is essentially the same).
Then is increasing on , and
is decreasing on . By (15.30), for all
, and for all
.
We have
and
It follows that
15.33
Example.
When you look at the graph of a function, you can usually ``see'' the
points where the second derivative changes sign. However, most people
cannot ``see'' points where the second derivative is undefined.
By inspecting graph
, you can see that
has a discontinuity at
.
By inspecting graph, you can see that is continuous everywhere,
but
is not defined at .
By inspecting graph in figure a below, you can see
that is continuous, but
you may have a hard time seeing the point where is not defined.
The function is defined by

(15.34) 
so
for
, and
for
, and
is not defined. We constructed
by
pasting together two parabolas. Figure b shows the two
parabolas, one having a second derivative equal to 1, and the other having
second derivative equal to 2.
15.35
Exercise.
Let
be the function described
in formula (
15.34). Draw graphs of
and
.
15.36
Entertainment (Discontinuous derivative problem.)
There exists a function
such that
is differentiable
everywhere on
, but
is discontinuous somewhere. Find such a function.
15.37
Exercise.
Let
. Show that
, but
is not
a point of inflection for
. Explain why this result does not contradict
theorem
15.32
15.38
Example.
Let
Then
and
Thus the only critical point for
is
. Also,
so
is increasing on
and is decreasing on
.
Thus
has a maximum at
, and
has no minima.
We see that
, and moreover
so
spills water over the interval
,
and
holds water over each of the intervals
and
.
Thus
has points of inflection at
.
We can use all of this information to make a
reasonable sketch of the graph of
.
Note that
for all
,
, and
, and
is approximately 0.58.
15.39
Exercise.
Discuss the graphs of the following functions. Make
use of all the information that you can get by looking at the functions
and their first two derivatives.
a)
.
b)
.
c)
.
Next: 16. Fundamental Theorem of
Up: 15. The Second Derivative
Previous: 15.2 Acceleration
Index
Ray Mayer
20070907