Verify Rolle’s theorem for each of the following functions:
Condition (1):
Since, f(x)=sin3x is a trigonometric function and we know every trigonometric function is continuous.
⇒ f(x)= sin3x is continuous on [0,π].
Condition (2):
Here, f’(x)= 3cos3x which exist in [0,π].
So, f(x)= sin3x is differentiable on (0,π).
Condition (3):
Here, f(0)=sin0=0
And f(π)=sin3π=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. 3cos3c =0
i.e.
i.e.
Value of
Thus, Rolle’s theorem is satisfied.