Q. 1 A4.7( 3 Votes )

# Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = 3 + (x – 2)^{2/3} on [1, 3]

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.

(i) Given function is:

⇒ on [1,3]

Let us check the differentiability of the function f(x).

Let’s find the derivative of f(x),

⇒

⇒

⇒

⇒

⇒

Let’s the differentiability at the value of x = 2

⇒

⇒

⇒

⇒

∴ f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).

So, Rolle’s theorem is not applicable for the function f on the interval [1,3].

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