#### 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] for x ∈ [– 2, 2]

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#### 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] for x ∈ [– 2, 2]

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

⇒ *f* (*x*) is not continuous in [−2, 2].

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#### Reference Material

Solution for 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] for X ∈ [– 2, 2] concept: Mean Value Theorem. For the courses CBSE (Arts), CBSE (Commerce), CBSE (Science)