Problem

Consider a continuous function f with the following properties:

1. f(–2) = 0
   
2. f(0) = 1
   
3. f(2) = 0
   
4. f has a local maximum at x = 0

Which of the following are possible graphs of f '(x)?

A.

B.

C.

D.

E.

Solution

First, note that we are looking for candidates for the graph of the derivative of f, not f itself, so we need to see what information about f ' we can gather from the properties of f.

Since f has a local maximum at x = 0, that means f '(0) = 0. This allows us to eliminate B and C.

Next, we think about how the values of f(x) change (since derivatives measure change!) as x goes from –2 to 2. At x = –2, f(x) = 0. At x = 0, f(x) = 1. Since f is continuous, f must increase somewhere between x = –2 and x = 0 to get from f(x) = 0 to f(x) = 1. Therefore, f ' must be positive somewhere in the interval [–2, 0]. (Not necessarily everywhere, since for all we know, f may also decrease somewhere in that interval. But we do know it must increase at least over some part of the interval.) We have a similar situation in the interval [0, 2]: f decreases, so f ' must be negative somewhere in this interval.

This information allows us to eliminate choice E. Once we have considered all the properties of f we have been given, we cannot eliminate further graphs. Therefore both A and D are possible graphs of the derivative of f.