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?

1. f has an inflection point at x = -1.5.
2. f is increasing on the interval from x = -3.2 to x = -4.5.
3. f has a relative minimum at x = 1.5.
4. All of these are true.

An inflection point is a point in which the function f(x) changes concavity: this means that the first derivative of a function has a relative minimum or maximum, and therefore the second derivative is zero, [tex]f''(x_0)=0[/tex].

In this case, we see that at the point

x = -1.5

The first derivative is flat: this means that at this point, the second derivative is zero, so this is an inflection point.

2. f is increasing on the interval from x = -3.2 to x = -4.5. --> FALSE.

We simply don't know this: in fact, we cannot see the graph of the derivative between -4.5 and -3.2, therefore we don't know if the function is increasing or not.

3. f has a relative minimum at x = 1.5 --> FALSE.

A function [tex]f(x)[/tex] has a relative minimum at a point [tex]x_0[/tex] if the first derivative [tex]f'(x)[/tex] is zero at that point:

[tex]f'(x_0)=0[/tex]

and moreover, the derivative is negative on the left of the point and positive on the right of the point.

This is not the case: in fact, we see that at x = 1.5 the first derivative is not zero, therefore this statement is not true.