Question

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

f(x) = e^-x sin x in [0, π]

Answer 1

Condition (1):

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

⇒ f(x) = e^-x sin x is continuous on [0, π].

Condition (2):

Here, f’(x) = e^-x (cos x – sin x) which exist in [0,π].

So, f(x) = e^-x sin x is differentiable on (0,π)

Condition (3):

Here, f(0) = e^-0 sin0 = 0

And f(π) = e^-πsinπ = 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. e^-c (cos c – sin c) = 0

i.e. cos c - sin c = 0

i.e. c = π/4

Value of c = π/4 ∈ (0, π)

Thus, Rolle’s theorem is satisfied.