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) = sinx + cosx on

We know that sine and cosine functions are continuous and differentiable on R.

Let’s the value of function f at extremums:

⇒ f(0) = sin(0) + cos(0)

⇒ f(0) = 0 + 1

⇒ f(0) = 1

⇒

⇒

⇒

We have got . So, there exists a cϵ such that f’(c) = 0.

Let’s find the derivative of the function ‘f’.

⇒

⇒ f’(x) = cosx – sinx

We have f’(c) = 0

⇒ cosc – sinc = 0

⇒

⇒

⇒

⇒

⇒

∴ Rolle’s theorem is verified.