#### Question

Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?

f (x) = x^{2} – 1 for x ∈ [1, 2]

#### Solution

By Rolle’s Theorem, for a function f: [a, b] → R, if

(a) f is continuous on [a, b]

(b) f is differentiable on (a, b)

(c) f (a) = f (b)

then, there exists some c ∈ (a, b) such that f'(c) = 0

Therefore, Rolle’s Theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis.

f (x) = x^{2} – 1 for x ∈ [1, 2]

It is evident that *f*, being a polynomial function, is continuous in [1, 2] and is differentiable in (1, 2).

Is there an error in this question or solution?

Solution Examine If Rolle’S Theorem is Applicable to Any of the Following Functions. Can You Say Some Thing About the Converse of Rolle’S Theorem from These Examples? F (X) = X2 – 1 for X ∈ [1, 2] Concept: Mean Value Theorem.