Given that f ' (3)=0 and f "(x)>0 for x(-5,5), what do you know about the function? What do you not know?

Question

calculus question regarding understanding derivatives

Answer 1

These are the givens:

A) f'(3) = 0

B) f"(x) > 0 on the interval (-5,5)

1) We do not know anything about the function outside the interval (-5,5).

The next statements refer only to the interval (-5,5).

2) From B we know that the function and its first two derivatives are defined.

Also the function and the first derivative are continuous and differentiable.

3) From B we know that the function is convex up.

4) From A and B we know that the function has a minimum at x = 3.

5) We do not know anything about the values of f(x).

6) We do not know anything about the values of f"(x) other than B.

7) From A and B we know that f'(x) < 0 on the interval (-5, 3) and

f'(x) > 0 on the interval (3, 5).