Let f : R R be defined as follows: f(x) = ≤ft(|x|/2 - x)≤ft(x - |x|/2) Which of the following statements is/are true?

f' is not differentiable over R

GATE CSE 2026 Session 1 Q46: Extrema and smoothness of (|x|/2 − x)(x − |x|/2)

Let $f : \mathbb{R} \to \mathbb{R}$ be defined as follows:

$f(x) = \left(\frac{|x|}{2} - x\right)\left(x - \frac{|x|}{2}\right)$

Which of the following statements is/are true?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

Show answer and explanation

Answer: A, C, D (official GATE 2026 final answer key)

Concept used: Split at $x = 0$ to remove the absolute value, then compare one-sided derivatives.

First step: For $x \ge 0$ , $f(x) = -x^2/4$ ; for $x < 0$ , $f(x) = -9x^2/4$ .

Detailed working, common mistakes and linked mock tests will be available with a free GATEiit account. Log-in is coming soon.

Share this question

Source: official GATE 2026 Session 1 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key