If ƒ(x) = |x| r , where a i ′ s are real constants, then ƒ…

Maths Functions, Limits, Continuity and Differentiability General Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 13, 2026

If ƒ(x) = |x| r , where a i ′ s are real constants, then ƒ(x) is-

Continuous at x = 0 for all a i

Differentiable at x = 0 for all a i ∈ R

Differentiable at x = 0 for all a 2k + 1 = 0

None of these

Text Solution

Verified by Experts

The correct answer is:

(a, c)

We know that | x | r , r = 0, 1, 2,…. are all continuous every where.

∴ ƒ(x) = | x | r is every where continuous

Since | x |, | x | 3 , | x | 5 , … are not differentiable at x = 0.

Whereas | x | 2 , | x | 4 , … are every where differentiable.

∴ ƒ(x) = | x | r is not differentiable at x = 0, if any one of a 1 , a 3 , a 5 , … is non-zero

Thus, for ƒ(x) to be differentiable at x = 0, we must have

a 1 = a 3 = a 5 …. = 0

i.e., a 2k + 1 = 0.

Hence, and are correct answer.

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