calculus question

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 and a relative maximum at x = -1.5 and a relative minimum at x = 1.5. Which of the following statement is false?

	f is concave up from x = -1.5 to x = 1.5.
	f has an inflection point at x = 1.5.
	f has a relative minimum at x = 2.
	All of these is false.

Answer 1

First, f is concave down from x=-1.5 to 1.5, since the slope (shown by f') is decreasing for that interval. So the first choice is false.

Since the slope hits a relative minimum at x=1.5 with the slope decreasing on the interval (1.5,1.5) and increasing for x>1.5, there is indeed an inflection point there. So the second choice is true.

Finally, since the slope is negative as x approaches 2 from the left and positive from the right, we see that f does have a relative minimum at x=2. So the third choice is true.

Since the second and third choices are true, not all choices are false. So the correct answer is the first choice.