## The Second Derivative and the Shape of a Function f(x)

Read this section to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1-9.

### Concavity

Graphically, a function is concave up if its graph is curved with the opening upward (Fig. 1a). Similarly, a function is concave down if its graph opens downward (Fig. 1b). The concavity of a function can be important in applied problems and can even affect billion-dollar decisions.

**Fig. 1**

An Epidemic: Suppose an epidemic has started, and you, as a member of congress, must decide whether the current methods are effectively fighting the spread of the disease or whether more drastic measures and more money are needed. In Fig. 2, is the number of people who have the disease at time , and two different situations are shown. In both and , the number of people with the disease, now), and the rate at which new people are getting sick, (now), are the same. The difference in the two situations is the concavity of , and that difference in concavity might have a big effect on your decision. In , is concave down at "now", and it appears that the current methods are starting to bring the epidemic under control. In , is concave up, and it appears that the epidemic is still out of control.

**Fig. 2**

Usually it is easy to determine the concavity of a function by examining its graph, but we also need a definition which does not require that we have a graph of the function, a definition we can apply to a function described by a formula without having
to graph the function.

**Definition:** Let be a differentiable function.

is **concave up** at a if the graph of is above the tangent line to for all close

is concave down at if the graph of is below the tangent line to for all close

Fig. 3 shows the concavity of a function at several points. The next theorem gives an easily applied test for the concavity of a function given by a formula.

**Fig. 3**

**The Second Derivative Condition for Concavity**

(a) If on an interval , then is increasing on and is concave up
on .

(b) If on an interval , then is decreasing on and is concave down on .

(c) If , then may be concave up or concave down or neither at .

Proof: (a) Assume that for all in , and let a be any point in . We want to show that is concave up at so we need to prove that the graph of
(Fig. 4) is above the tangent line to at : for close to .

**Fig. 4**

Assume that is in , and apply the Mean Value Theorem to on the interval from to . Then there is a number between and so that

Since between and , we know from the Second Shape Theorem that is increasing between and . We will consider two cases: and .

If , then and is in the interval so . Since is increasing, implies that . Multiplying each side
of the inequality by the positive amount , we get that .
Adding to each side of this last inequality, we have .

If , then and is in the interval so . Since is increasing, a implies that . Multiplying each side
of the inequality by the **negative** amount , we get that
and .

In each case we get that the function is above the tangent line . The proof of is similar.

(c) Let , and (Fig.5). The second derivative of each of these functions is zero at , and at they all have the same tangent line: , the -axis. However, at they all have
different concavity: is concave up, is concave down, and is neither concave up nor down.

**Fig. 5**

**Practice 1:** Use the graph of in Fig. 6 to finish filling in the table with "" for positive, "" for negative" or "".

**Fig. 6**