The graph of f ′(x) is continuous and increasing with an x-intercept at x = 5. Which of the following statements must be true?

Question

The graph of f ′(x) is continuous and increasing with an x-intercept at x = 5. Which of the following statements must be true?

 The graph of f is always concave up.

 The graph of f has an inflection point at x = 5. 

The graph of f has a relative maximum at x = 5. 

None of these is true.

Answer 1

Rachel:

f'(x) > 0 for x>5 and f'(x) < 0 for x<5. So f(x) is decreasing on x<5 and increasing on x>5. Obviously, it can't have a relative maximum at x=5.

Since f'(x) is increasing, the slope of f'(x) is always positive. The slope of f'(x) is f ''(x). So f ''(x) > 0 and f is always concave up.