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# 15.3 Convexity

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 2007-09-07