Question

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

f(x) = 4^{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:

⇒ f(x) = 4^sinx on [0, π]

We that sine function is continuous and differentiable over R.

Let’s check the values of function ‘f’ at extremums

⇒ f(0) = 4^sin(0)

⇒ f(0) = 4⁰

⇒ f(0) = 1

⇒ f(0) = 4^sinπ

⇒ f(π) = 4⁰

⇒ f(π) = 1

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

Let’s find the derivative of ‘f’

∴ Rolle’s theorem is verified.