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

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?

 The graph of f is always concave down. 

The graph of f is always increasing. 

The graph of f has a relative maximum at x = 0. 

The graph of f has a relative minimum at x = 0.

Answer 1

Everywhere that f'(x) is positive, f is increasing. Since f'(x) is positive everywhere, f(x) must be increasing everywhere.