Question

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

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

Answer 1

Answer: (D)

Let $f(x)=\sin (|x|)-|x|$
$\Rightarrow f\left(x\right)=\{\begin{array}{l}\sin x-x,x>0\\ -\sin x+x,x>0\end{array}$
$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{l}\cos 1-x,x\ge 0\\ -\cos x+1,x>0\end{array}$
$\therefore$ L.H.D = ${\lim }_{x\to 0^{-}}\left(-\cos x+1\right)=-1+1=0$
R.H.D = ${\lim }_{x\to 0^{+}}\left(\cos x-1\right)=-1-1=0$
$\because$ L.H.D = R.H.D
$\therefore f(x)$ is differentiable at $x=0$