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

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

$=\left\{\begin{array}{cc}sinx-x,& x\ge 0\\ -sinx+x,& x<0\end{array}\right\}$

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

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