Verify Rolle’s theorem for each of the following functions:
Condition (1):
Since, f(x)=cosx is a trigonometric function and we know every trigonometric function is continuous.
⇒ f(x)=cosx is continuous on .
Condition (2):
Here, f’(x)=-sinx which exist in .
So, f(x)=cosx is differentiable on .
Condition (3):
Here,
And
i.e.
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one such that f’(c)=0
i.e. -sinc=0
i.e. c=0
Value of
Thus, Rolle’s theorem is satisfied.