Question

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

• A. $\cos \left(|x|\right)+|x|$
• B. $\cos \left(|x|\right)-|x|$
• C. $\sin \left(|x|\right)+|x|$
• D. $\sin \left(|x|\right)-|x|$

Answer 1

The correct option is D $\sin \left(|x|\right)-|x|$

$g\left(x\right)=\cos \left(|x|\right)\pm |x|=\cos \left(x\right)\pm |x|(\because \text{cos x is an even function})$
$g\left(x\right)$ is not differentiable at $x=0$

$f\left(x\right)=\sin \left(|x|\right)-|x|$

$f\left(x\right)=\left\{\begin{array}{c}\sin x-x,x\ge 0\\ -\sin x+x,x<0\end{array}\right\}$

$f^{{}^{\prime }}\left(0^{+}\right)=0=f^{{}^{\prime }}\left(0^{-}\right)$

So, $f\left(x\right)$ is differentiable at $x=0$.

Similarly, check for $\sin \left(|x|\right)+|x|$