Verify Rolle’s theorem for each of the following functions on the indicated intervals :

Question

Philoid · Accepted Answer

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) = sin⁴x + cos⁴x on

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

Let’s find the value of function ‘f’ at extremums

⇒ f(0) = sin⁴(0) + cos⁴(0)

⇒ f(0) = (0)⁴ + (1)⁴

⇒ f(0) = 0 + 1

⇒ f(0) = 1

⇒

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

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

⇒

⇒ f’(x) = 4sin³xcosx–4cos³xsinx

⇒ f’(x) = 4sinxcosx(sin²x – cos²x)

⇒ f’(x) = 2(2sinxcosx)( – cos2x)

⇒ f’(x) = – 2(sin2x)(cos2x)

⇒ f’(x) = – sin4x

We have f’(c) = 0

⇒ – sin4c = 0

⇒ sin4c = 0

⇒ 4c = 0 or

⇒

∴ Rolle’s theorem is verified.