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) = x² – 1 for x ∈ [1, 2]

Answer 1

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) = x² – 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).