First, let us write the conditions for the applicability of Rolle’s theorem:

For a Real valued function ‘f’:

a) The function ‘f’ needs to be continuous in the closed interval [a,b].

b) The function ‘f’ needs differentiable on the open interval (a,b).

c) f(a) = f(b)

Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.

Given function is :

⇒

We know that cosine function is continuous and differentiable on R.

Let’s find the values of the function at an extremum,

⇒

⇒

⇒

We know that cos( – x) = cosx

⇒ f(0) = 0

⇒

⇒

⇒

⇒

We got , so there exist a such that f’(c) = 0.

Let’s find the derivative of f(x)

⇒

⇒

⇒

We have f’(c) = 0,

⇒

⇒

⇒

∴ Rolle’s theorem is verified.