Analyzing Functions

Question

Given the graph of f '(x), the derivative of f(x), which of the following statements are true about the graph of f(x)?

I. The graph of f has a critical point at x = 0.

II. The graph of f has a critical point at x = 1.

III. The graph of f has a point of inflection at x = 1.

Options:

1. I only
2. II only
3. III only
4. I and III only

Answer 1

I. The graph of f has a critical point at x = 0.

Since critical points occur when f'(x) = 0 or f'(x) = Does not exist, at x = 0 on your graph above f'(x) = 0 so this is a critical point and the statement is True

II. The graph of f has a critical point at x = 1.

Since f'(x) is not 0 at x = 1 and it does exist at that point, x = 1 is not a critical point. This statement is false.

III. The graph of f has a point of inflection at x = 1.

An inflection point occurs where f''(x) = 0. f''(x) is the slope of f'(x) graphed above. Notice the slope is zero at x = 1 for the graph above. So f''(x) = 0 at x = 1. There is an inflection point x = 1.

So the answer is I and III only are true (Choice #4)