Q. 3 O5.0( 2 Votes )

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

f(x) = 4^{sin x} on [0, π]

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) = 4^{sinx} on [0,]

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) = 4^{0}

⇒ f(0) = 1

⇒ f() = 4^{sinπ}

⇒ f() = 4^{0}

⇒ f() = 1

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

Let’s find the derivative of ‘f’

⇒

⇒

⇒

We have f’(c) = 0

⇒ 4^{sinc}log4cosc = 0

⇒ cosc = 0

⇒

∴ Rolle’s theorem is verified.

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