Please help and thank you!

Question

Which of the following statements is/are true? (5 points)

I. If f '(c) = 0, then f has a local maximum or minimum at x = c.

II. If f is continuous on [a, b] and differentiable on (a, b) and f '(x) = 0 on (a, b), then f is constant on [a, b].

III. The Mean Value Theorem can be applied to f(x) = 1/x² on the interval [−1, 1].

Options:

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

Answer 1

Let's examine the statements:

I. If f '(c) = 0, then f has a local maximum or minimum at x = c.

f'(c) = 0 means that the slope of f(x) = 0 at x = c, so yes f has a local maximum or minimum at x = c...TRUE

II. If f is continuous on [a, b] and differentiable on (a, b) and f '(x) = 0 on (a, b), then f is constant on [a, b].

If f'(x) = 0 on (a, b), then f is neither increasing nor decreasing and thus is constant on [a, b]...TRUE

III. The Mean Value Theorem can be applied to f(x) = 1/x² on the interval [−1, 1].

No, f is neither continuous nor differentiable at x = 0...FALSE