need help with a calculus problem !

Question

Below is the graph of f ′(x), the derivative of f(x), and has x-intercepts at x = -3, x = 1 and x = 2. There are horizontal tangents at x = -1.5 and x = 1.5. Which of the following statements is true?

A) f has an inflection point in the interval x = -1 to x = 1.

B) f is increasing on the interval from x = -3 to x = 1.

C) f has a relative maximum at x = -1.5.

D) None of these is true.

Answer 1

In general,

f has a relative minimum or maximum where f' is 0, i.e., at x = -3, 1, 2

f has an inflection point when f" is 0, i.e., at x = -1.5, 1.5

All three statements are false.

A) f does not have an inflection point in the interval x = -1 to x = 1, because

neither of the two inflection points -1.5, 1.5 lies in that interval.

B) f is decreasing on the interval from x = -3 to x = 1, because

f' is negative there.

C) f does not have a relative maximum at x = -1.5, because f'(-1.5) ≠ 0.