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 :



We know that cosine function is continuous and differentiable on R.


Let’s find the values of the function at an extremum,





We know that cos( – x) = cosx


f(0) = 0






We got , so there exist a such that f’(c) = 0.


Let’s find the derivative of f(x)





We have f’(c) = 0,





Rolle’s theorem is verified.


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