Question
Which of the following functions is differentiable at $$x = 0? $$
A.
$$\cos \,\left( {\left| x \right|} \right) + \left| x \right|$$
B.
$$\cos \,\left( {\left| x \right|} \right) - \left| x \right|$$
C.
$$\sin \,\left( {\left| x \right|} \right) + \left| x \right|$$
D.
$$\sin \,\left( {\left| x \right|} \right) - \left| x \right|$$
Answer :
$$\sin \,\left( {\left| x \right|} \right) - \left| x \right|$$
Solution :
Let us test each of four options.
\[\begin{array}{l}
\left( {\rm{A}} \right)f\left( x \right) = \cos \,\left( {\left| x \right|} \right) + \left| x \right| = \left\{ \begin{array}{l}
\cos \,x - x,\,\,x < 0\\
\cos \,x + x,\,\,x \ge 0
\end{array} \right.\\
f'\left( x \right) = \left\{ \begin{array}{l}
- \sin \,x - 1,\,x < 0\\
- \sin \,x + 1,\,\,x \ge 0
\end{array} \right.\\
{\rm{At}}\,{\rm{ }}x = 0,\,\,LD = - 1,\,\,RD = 1\\
\therefore {\rm{Not\,differentiable }}\\
\left( {\rm{B}} \right)f\left( x \right) = \cos \,\left| x \right| - \left| x \right| = \left\{ \begin{array}{l}
\cos \,x + x,\,\,x < 0\\
\cos \,x - x,\,\,x \ge 0
\end{array} \right.\\
\therefore {\rm{Not\,differentiable\,at }}\,x = 0\\
\left( {\rm{C}} \right)f\left( x \right) = \sin \,\left| x \right| + \left| x \right| = \left\{ \begin{array}{l}
- \sin \,x - x,\,\,x < 0\\
\sin \,x - x,\,\,x \ge 0
\end{array} \right.\\
\therefore {\rm{Not\,differentiable\,at }}\,x = 0\\
\left( {\rm{D}} \right)f\left( x \right) = \sin \,\left| x \right| - \left| x \right| = \left\{ \begin{array}{l}
- \sin \,x + x,\,\,x < 0\\
\sin \,x - x,\,\,x \ge 0
\end{array} \right.\\
f'\left( x \right) = \left\{ \begin{array}{l}
- \cos \,x - 1,\,\,x < 0\\
\cos \,x - 1,\,\,x \ge 0
\end{array} \right.\\
{\rm{At }}x = 0,\,\,LD = 0,\,\,RD = 0\,\\
\therefore f\,{\rm{ is\,differentiable\,at }}\,x = 0
\end{array}\]