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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