Question

Can Rolle's theorem be applied to f on the closed interval [a, b]?
1) Yes
2) No

Answer 1

Rolle's theorem cannot be applied to f on the closed interval [a, b] because the function needs to be differentiable on the open interval (a, b).

Step-by-step explanation:

Rolle's theorem states that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.

However, Rolle's theorem cannot be applied to f on the closed interval [a, b] because the theorem requires the function to be differentiable on the open interval (a, b). Since the interval in this question is closed, it means that the endpoints a and b are included, making the function not differentiable at those points.

Therefore, the answer to whether Rolle's theorem can be applied to f on the closed interval [a, b] is No (option 2).