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

Published by: CGP EDU Academic Team

Published on: August 13, 2026

For ƒ(x) = x 2 – 2 |x|, test the continuity and differentiability of g(x) in the interval [–2, 3], where

g (x) = , are -

g (x) is continuous and differentiable everywhere except x = 0

g (x) is non-differentiable at x = 0 and 2

g (x) is non-differentiable at x = 0, 1, 2

None of the above

Text Solution

Verified by Experts

The correct answer is:

Here ƒ(x) =

Case 1 : –2 ≤ x < –1, ƒ(x) decreases ⇒ g (x) = x 2 + 2x

Case 2 : –1 ≤ x < 0, ƒ(x) increases ⇒ g (x) = –1

Case 3 : 0 ≤ x < 1, ƒ(x) decreases ⇒ g(x) = 0

Case 4 : 1 ≤ x < 2, ƒ(x) increases but ƒ(x) < 0

⇒ g (x) = ƒ(0) = 0

Case 5 : 2 ≤ x ≤ 3, ƒ(x) increases and ƒ(x) ≥ 0

∀ x ∈ [2, 3)

⇒ g (x) = ƒ (x) = x 2 – 2x, 2 ≤ x ≤ 3

Therefore g (x) =

Clearly g (x) is continuous everywhere except at x = 0.

Also g (x) is non differentiable at x = 0 and 2.

Hence is the correct answer.

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