​ Please solve

​ Please solve Which of the following function is differentiable at x = 0? (a) cos(lxl) + IXI (b) cos(lxl) — IXI (c) sin(lxl) + IXI (d) sin(lxl) — lx

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$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\left|\mathrm{x}\right|\right)-\left|\mathrm{x}\right| \mathrm{is} \mathrm{differentiable} \mathrm{at} \mathrm{x}=0. \mathrm{Let} \mathrm{us} \mathrm{see} \mathrm{how} \\ \mathrm{f}\left(\mathrm{x}\right)=\sin \left(\left|\mathrm{x}\right|\right)-\left|\mathrm{x}\right| \\ \mathrm{f}\left(0\right)=0 \\ \mathrm{Any} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{derivable} \mathrm{at} \mathrm{x}=\mathrm{a} \mathrm{if} \\ \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{a}\right)}{\mathrm{x}-\mathrm{a}} \mathrm{exists} \\ \mathrm{Right} \mathrm{hand} \mathrm{derivative} \mathrm{is} \mathrm{given} \mathrm{by} \\ \mathrm{R}.\mathrm{H}.\mathrm{D}.=\underset{\mathrm{x}\to 0^{+}}{\lim } \frac{\sin \left|\mathrm{x}\right|-\left|\mathrm{x}\right|-\mathrm{f}\left(0\right)}{\mathrm{x}-0} \\ \underset{\mathrm{x}\to 0^{+}}{\lim } \frac{\sin \left|\mathrm{x}\right|-\left|\mathrm{x}\right|}{\mathrm{x}} \\ \mathrm{For} \mathrm{x}>0, \mathrm{mod} \mathrm{will} \mathrm{open} \mathrm{with} \mathrm{negatibe} \mathrm{sign} \\ =\underset{\mathrm{x}\to 0^{+}}{\lim } \frac{\mathrm{sinx}-\mathrm{x}}{\mathrm{x}} \\ \mathrm{Apply} \mathrm{L}-\mathrm{H} \mathrm{rule} \\ =\underset{\mathrm{x}\to 0^{+}}{\lim } \frac{\mathrm{cosx}-1}{1} \\ =\frac{\cos 0-1}{1} \\ =1-1 \\ =0 \\ \mathrm{Left} \mathrm{hand} \mathrm{derivative} \mathrm{is} \mathrm{given} \mathrm{by} \\ \mathrm{L}.\mathrm{H}.\mathrm{D}.=\underset{\mathrm{x}\to 0^{+}}{\lim } \frac{\sin \left|\mathrm{x}\right|-\left|\mathrm{x}\right|-\mathrm{f}\left(0\right)}{\mathrm{x}-0} \\ \underset{\mathrm{x}\to 0^{-}}{\lim } \frac{\sin \left|\mathrm{x}\right|-\left|\mathrm{x}\right|}{\mathrm{x}} \\ \mathrm{For} \mathrm{x}<0, \mathrm{mod} \mathrm{will} \mathrm{open} \mathrm{with} \mathrm{negatibe} \mathrm{sign} \\ =\underset{\mathrm{x}\to 0^{-}}{\lim } \frac{-\mathrm{sinx}+\mathrm{x}}{\mathrm{x}} \\ \mathrm{Apply} \mathrm{L}-\mathrm{H} \mathrm{rule} \\ =\underset{\mathrm{x}\to 0^{-}}{\lim } \frac{-\mathrm{cosx}+1}{1} \\ =\frac{-\cos 0+1}{1} \\ =-1+1 \\ =0 \\ \mathrm{As} \mathrm{L}.\mathrm{H}.\mathrm{D}.=\mathrm{R}.\mathrm{H}.\mathrm{D}. \mathrm{hence} \mathrm{function} \mathrm{is} \mathrm{derivable} \mathrm{at} \mathrm{x}=0 \end{gathered}$$

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