Question

Verify Rolle’s theorem for the following functions.

f(x) = sin(x/2), x ∈ [0, 2π]

Answer 1

The function f(x) = sin(x/2) is continuous on [0, 2π] and differentiable on (0, 2π).

Now, f(0) = sin 0 = 0 

and f(2π) = sin π = 0 

∴ f(0) = f(2π)

Thus, the function f satisfies all the conditions of Rolle’s theorem. 

∴ there exists c ∈ (0, 2π) such that f'(c) = 0.

Hence, Rolle’s theorem is verified.