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, π]
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(π) = cos2(π)
⇒ f(π) = cos(2π)
⇒ f(π) = 1
We got f(0) = f(π), so there exist a c∈(0, π) such that f’(c) = 0.
Let’s find the derivative of f(x)
∴ Rolle’s theorem is verified.