Verify Rolle’s theorem for each of the following functions:
Condition (1):
Since, f(x)=sinx+cosx is a trigonometric function and we know every trigonometric function is continuous.
⇒ f(x)= sinx+cosx is continuous on .
Condition (2):
Here, f’(x)= cosx-sinx which exist in .
So, f(x)= sinx+cosx is differentiable on
Condition (3):
Here, f(0)=sin0+cos0=1
And
i.e.
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one such that f’(c)=0
i.e. cosc-sinc =0
i.e.
Value of
Thus, Rolle’s theorem is satisfied.