Answer :

Condition (1):


Since, f(x)=e-x sinx is a combination of exponential and trigonometric function which is continuous.


f(x)= e-x sinx is continuous on [0,π].


Condition (2):


Here, f’(x)= e-x (cosx – sinx) which exist in [0,π].


So, f(x)= e-x sinx is differentiable on (0,π)


Condition (3):


Here, f(0)= e-0 sin0=0


And f(π)= esinπ =0


i.e. f(0)=f(π)


Conditions of Rolle’s theorem are satisfied.


Hence, there exist at least one cϵ(0,π) such that f’(c)=0


i.e. e-c (cos c – sin c) =0


i.e. cos c-sin c = 0


i.e.


Value of


Thus, Rolle’s theorem is satisfied.


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