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GATE 2022 AE – Question 43

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 2 marks · Multiple select

The real function $y=\sin^2(|x|)$ is

  1. continuous for all x
  2. differentiable for all x
  3. not continuous at x=0
  4. not differentiable at x=0

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Correct answer: (A) continuous for all x; (B) differentiable for all x

Explanation

**Simplify the function.** For any real $x$, $\sin^2(|x|)=\left(\sin|x|\right)^2$. Since $\sin(-x)=-\sin x$, we have $\sin^2(|x|)=\sin^2x$ for all real $x$ (the sign disappears when squared).

So $y=\sin^2x$, a composition of smooth functions:
- It is continuous for all $x$. ✓ (A)
- Its derivative $2\sin x\cos x=\sin2x$ exists for all $x$, including $x=0$, so it is differentiable everywhere. ✓ (B)

Statements C and D are therefore false. (Compare $\sin|x|$ itself, which has a corner at the origin.)

Answer **A and B**.