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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