Question

Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?

Answer 1

By Rolle’s theorem, for a function, if

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

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

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

then there exists some c ∈ (a, b) such that .

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

(i)

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at x = 5 and x = 9.

Thus, f (x) is not continuous on [5, 9].

The differentiability of f on (5, 9) is checked in the following way.

Let n be an integer such that n ∈ (5, 9).

Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.

Thus, f is not differentiable on (5, 9).

It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.

Hence, Rolle’s theorem is not applicable on.

(ii)

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at x = −2 and x = 2.

Thus, f (x) is not continuous on [−2, 2].

The differentiability of f on (−2, 2) is checked in the following way.

Let n be an integer such that n ∈ (−2, 2).

Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.

Thus, f is not differentiable on (−2, 2).

It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.

Hence, Rolle’s theorem is not applicable on.