Q. 14

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



Answer :

Condition (1):


Since, f(x)=cos2x is a trigonometric function and we know every trigonometric function is continuous.


f(x)= cos2x is continuous on [0,π].


Condition (2):


Here, f’(x)= -2sin2x which exist in [0,π].


So, f(x)=cos2x is differentiable on (0,π).


Condition (3):


Here, f(0)=cos0=1


And f(π)=cos2π=1


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. -2sin2c =0


i.e. 2c=π


i.e.


Value of


Thus, Rolle’s theorem is satisfied.


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