Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = sin 3x on [0, π]

Question

Verify Rolle’s theorem for each of the following functions on the indicated intervals : f(x) = sin 3x on [0, π]

Philoid · Accepted Answer

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) = sin3x on [0,]

We know that sine function is continuous and differentiable on R.

Let’s find the values of function at extremum,

⇒ f(0) = sin3(0)

⇒ f(0) = sin0

⇒ f(0) = 0

⇒ f() = sin3()

⇒ f() = sin(3)

⇒ f() = 0

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

Let’s find the derivative of f(x)

⇒

⇒ f’(x) = 3cos3x

We have f’(c) = 0,

⇒ 3cos3c = 0

⇒

∴ Rolle’s theorem is verified.

Verify Rolle’s theorem for each of the following functions on the indicated intervals :f(x) = sin 3x on [0, π]