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.

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