Aspire Faculty ID #16729 · Topic: CUET 2023 · Just now
CUET 2023

Which of the following functions is differentiable at $x=0$?

Solution

$|x|$ is not differentiable at $0$.

Near $x=0$, $\sin(|x|)\approx |x|$.

For option (4):
$f(x)=\sin(|x|)-|x|$

Near $0$:
$f(x)\approx |x|-|x|=0$

Both left-hand and right-hand derivatives at $0$ are equal, hence $f(x)$ is differentiable at $0$.

All other options involve $|x|$ in a way that makes the derivative discontinuous at $0$.

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