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:

⇒ f(x) = cos 2x on [0,]

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

Let’s find the values of function at extremum,

⇒ f(0) = cos2(0)

⇒ f(0) = cos(0)

⇒ f(0) = 1

⇒ f() = cos2()

⇒ f() = cos(2)

⇒ f() = 1

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

Let’s find the derivative of f(x)

⇒

⇒

⇒ f’(x) = – 2sin2x

We have f’(c) = 0,

⇒ – 2sin2c = 0

⇒ 2c = 0

⇒

∴ Rolle’s theorem is verified.