Check the validity of the Rolle’s theorem for the following function

1.	Check the validity of the Rolle’s theorem for the following functions.f(x) = e-x sin x, x ∈ [0, π].
Answer» The functions e^-x and sin x are continuous and differentiable on their domains. ∴ f(x) = e^-x sin x is continuous on [0, π] and differentiable on (0, π). Now, f(0) = e⁰ sin 0 = 1 × 0 = 0 and f(π) = e^-π sin π = e^-π × 0 = 0 ∴ f(0) = f(π) Thus, the function f satisfies all the conditions of the Rolle’s theorem.

Check the validity of the Rolle’s theorem for the following functions.f(x) = e-x sin x, x ∈ [0, π].

Answer»

The functions e^-x and sin x are continuous and differentiable on their domains.

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

Now, f(0) = e⁰ sin 0 = 1 × 0 = 0

and f(π) = e^-π sin π = e^-π × 0 = 0

∴ f(0) = f(π)

Thus, the function f satisfies all the conditions of the Rolle’s theorem.