Question

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

f(x) = e^x sin x on [0, π]

Answer 1

First, let us write the conditions for the applicability of Rolle’s theorem:

For a Real valued function ‘f’:

a) The function ‘f’ needs to be continuous in the closed interval [a, b].

b) The function ‘f’ needs differentiable on the open interval (a, b).

c) f(a) = f(b) Then there exists at least one c in the open interval (a, b) such that f’(c) = 0.

Given function is:

\(\Rightarrow f(x)=e^xsin\,x\) on [0, π]

We know that exponential and sine functions are continuous and differentiable on R.

Let’s find the values of the function at an extremum,

⇒ f(0) = e⁰sin(0)

⇒ f(0) = 1 × 0

⇒ f(0) = 0

We got f(0) = f(π), so there exist a c∈(0, π) such that f’(c) = 0.

Let’s find the derivative of f(x)

∴ Rolle’s theorem is verified.