MathsJEE Main 8 Apr 2026, Shift 2·Q13·MCQ·Moderate

Limits, Continuity & Differentiability · Differentiability of modulus functions

For the function $f(x) = e^{\sin\lvert x\rvert} - \lvert x\rvert$ , $x \in \mathbb{R}$ , consider the following statements:

Statement I: $f$ is differentiable for all $x \in \mathbb{R}$ .

Statement II: $f$ is increasing in $\left(-\pi, -\dfrac{\pi}{2}\right)$ .

In the light of the above statements, choose the correct answer from the options given below:

(1)Both Statement I and Statement II are true
(2)Both Statement I and Statement II are false
(3)Statement I is true but Statement II is false
(4)Statement I is false but Statement II is true

We corrected a printing error in the original paper

Answer corrected; the source key is (4) — for $f(x) = e^{\sin\lvert x\rvert} - \lvert x\rvert$ both one-sided derivatives at $x = 0$ are $0$ and $f$ is smooth elsewhere, so Statement I is true, and $f'(x) = 1 - \cos x\, e^{-\sin x} > 0$ on $(-\pi, -\pi/2)$ , so Statement II is true; the source's solution works with $e^{\lvert\sin x\rvert}$ instead.

Video solution coming soon

We're animating every PYQ step by step. Subscribe to get them first.

See the full 8 Apr 2026, Shift 2 paper