Q. 14

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