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

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

For $x<0$

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

Derivating and putting $x=0$, we have

$f^{\prime }\left(x\right)=-\cos x+1$

$f^{\prime }\left(0\right)=-1+1=0$

For $x>0$

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

Derivating and putting $x=0$

$f^{\prime }\left(x\right)=\cos x-1$

$f^{\prime }\left(0\right)=1-1=0$

Derivatives from both the sides are equal.

So the function is differentiable.

Hence, D is correct.