Question

Determine whether Rolle's theorem can be applied to f on the closed interval [a, b] for the function f(x) = Bcos(πx), [0, 2].

Answer 1

Yes, Rolle's theorem can be applied to f(x) = Bcos(πx) on the closed interval [0, 2].

Step-by-step explanation:

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

In this case, the function f(x) = Bcos(πx) is continuous and differentiable on the interval [0, 2]. To determine if Rolle's theorem can be applied, we need to check if the conditions are met:

• f(x) = Bcos(πx) is continuous on [0, 2] as cosine function is continuous everywhere.
• f(x) = Bcos(πx) is differentiable on (0, 2) as cosine function is differentiable everywhere.
• f(0) = Bcos(0) = B and f(2) = Bcos(2π) = Bcos(0) = B.

Since f(a) = f(b), the conditions of Rolle's theorem are satisfied and we can conclude that there exists at least one value c in (0, 2) such that f'(c) = 0.