Question

Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = cos x, [π, 3π] Yes. No, because fis not continuous on the closed interval [a, b]. O No, because f is not differentiable in the open interval (a, b). No, because f(a) + f(b). > If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c)= 0.

Answer 1

Rolle's Theorem can be applied to f(x) = cos x on the closed interval [π, 3π]. Values of c in the open interval (a, b) such that f'(c) = 0 are x = nπ, where n is an integer.

Step-by-step explanation:

According to Rolle's Theorem, for a function to satisfy the conditions required for the theorem to be applicable on the closed interval [a, b], three conditions must be met:

1. The function must be continuous on the closed interval [a, b].
2. The function must be differentiable on the open interval (a, b).
3. The function must have the same value at the endpoints, meaning f(a) = f(b).

For the given function f(x) = cos x on the closed interval [π, 3π], all three conditions are satisfied. Therefore, Rolle's Theorem can be applied.

To find all values of c in the open interval (a, b) such that f'(c) = 0, we need to find where the derivative of cos x equals zero. Since the derivative of cos x is -sin x, setting -sin x = 0 gives us sin x = 0, and the solutions are x = nπ, where n is an integer.

Answer 2

Rolle's Theorem cannot be applied to f on the closed interval [a, b].

Step-by-step explanation:

To determine whether Rolle's Theorem can be applied to f on the closed interval [a, b], we need to check for the conditions of the theorem.

1. The function f(x) = cos x is continuous on the closed interval [π, 3π] as cosine function is continuous for all real numbers.
2. The function f(x) = cos x is differentiable in the open interval (π, 3π) as the derivative of cosine function exists everywhere.
3. f(a) = cos(π) = -1 and f(b) = cos(3π) = -1, so f(a) + f(b) = -1 + (-1) = -2, which is not greater than zero.

Therefore, Rolle's Theorem cannot be applied to f on the closed interval [a, b].