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Mathematics · Limits, Continuity and Differentiability

JEE Main 2026 — 8 April, Evening Shift — Question 30

For the function f(x)=esinx−∣x∣,x∈R,f(x)=e^{sin x} - |x|, x∈R, consider the following statements :

Statement I: f is differentiable for all x∈Rx∈R

Statement II: f is increasing in (−π,−π/2).(-π, -π/2).

  1. Option A:

    Both Statement I and Statement II are true

  2. Option B:

    Both Statement I and Statement II are false

  3. Option C:

    Statement I is true but Statement II is false

  4. Option D:

    Statement I is false but Statement II is true

    Correct

Answer: D

Step-by-step solution

f(x)=e∣sin⁡x∣−∣x∣f(x)=e^{|\sin x|}-|x| is non-differentiable at x=π\mathrm{x}=\pi and for x∈(−π,−π2)\mathrm{x} \in\left(-\pi, \frac{-\pi}{2}\right) f(x)=e−sinx+x\mathrm{f}(\mathrm{x})=\mathrm{e}^{-\mathrm{sin} \mathrm{x}}+\mathrm{x} f′(x)=−e−sin⁡xcos⁡x+1\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{e}^{-\sin \mathrm{x}} \cos \mathrm{x}+1 for x∈(−π,−π2)cos⁡x<0\mathrm{x} \in\left(-\pi, \frac{-\pi}{2}\right) \cos \mathrm{x}<0 ∴f′(x)>0\therefore \mathrm{f}^{\prime}(\mathrm{x})>0 ∴f(x)\therefore \mathrm{f}(\mathrm{x}) is increasing

Answer key and solution verified before publishing.

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Exam
JEE Main 2026
Paper
8 April, Evening Shift
Subject
Mathematics
Chapter
Limits, Continuity and Differentiability
Topic
Continuity

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