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

Condition (1):

Since, f(x)=e^-x (sinx-cosx) is a combination of exponential and trigonometric function which is continuous.

⇒ f(x)= e^-x (sinx-cosx) is continuous on .

Condition (2):

Here, f’(x)= e^-x (sinx + cosx) - e^-x (sinx – cosx)

= e^-x cosx which exist in .

So, f(x)= e^-x (sinx-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. e^-c cos c =0

i.e. cos c = 0

i.e.

Value of

Thus, Rolle’s theorem is satisfied.