Help with Calc please!

Question

The graph of f '(x) is continuous, positive, and has a relative maximum at x = 0. Which of the following statements must be true?

Options:

1)	The graph of f is always concave down.
2)	The graph of f is always increasing.
3)	The graph of f has a relative maximum at x = 0.
4)	The graph of f has a relative minimum at x = 0.

Answer 1

Has your teacher/professor talked about "sign" lines? It is kind of old-school, but they are very useful.

If f '(x) is continuous and positive, that means that the original function f(x) was continuously increasing.

If f '(x) has a relative max at x = 0, that makes it a "peak" - which makes it concave down, so the second derivative, f ''(x) will be negative at x=0.

Looking at your choices:

1) The graph of f(x) is always concave down. Well - it is concave down here surrounding x=0, but not necessarily everywhere. So we rule out answer #1.

2) The graph of f(x) is always increasing. If the first derivative is always positive then the original function f(x) is always increasing. This is probably the correct answer, but let us check the others to be sure.

3) The graph of f(x) has a relative maximum at x=0. This is very much NOT TRUE, the first derivative has a maximum at x=0, so f(x) is concave down near x=0, but if the original function f(x) is always increasing, it cannot have a maximum. (To have a maximum it would have to "turn over" and become decreasing). So we rule out #3

4) The graph of f(x) has a relative minimum at x=0. This is very much NOT TRUE, the first derivative has a maximum at x=0, so f(x) is concave down near x=0, but if the original function f(x) is always increasing, it cannot have a minimum. (To have a minimum it would have had to be decreasing, then "turn over" and become increasing). So we rule out #4

Correct answer is #2