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

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:


on [ – 1,1]


We know that exponential function is continuous and differentiable over R.


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




f( – 1) = e0


f( – 1) = 1




f(1) = e0


f(1) = 1


We got f( – 1) = f(1) so, there exists a cϵ( – 1,1) such that f’(c) = 0.


Let’s find the derivative of the function f:





We have f’(c) = 0



2c = 0


c = 0ϵ[ – 1,1]


Rolle’s theorem is verified.


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