Q102 / 160 · AP EAMCET 2022 (07 Jul Shift 1) · Mathematics

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

  • A
    $f(x)=\cos |x|+|x|$
  • B
    $f(x)=\sin |x|+|x|$
  • C
    $f(x)=\cos |x|-|x|$
  • Correct answer
    D
    $f(x)=\sin |x|-|x|$

Solution

At $x=0$, RHD $\sin |x|-|x|=\lim _{h \rightarrow 0} \frac{(\sin |h|-|h|-(0)}{h}$ $\begin{aligned} & =\lim _{h \rightarrow 0}\left(\frac{\sin h}{h}-1\right) \\ & =1-1=0\end{aligned}$ LHD of $\sin |x|-|x|=\lim _{h \rightarrow 0} \frac{(\sin |-h|-|-h|)}{-h}=0$ $\begin{aligned} & =\lim _{h \rightarrow 0}\left[\left(\frac{\sin h}{-h}\right)+1\right] \\ & =0\end{aligned}$ $L H D=R H D$ $\therefore \sin |x|-|x|$ is differentiable at $x=0$

Hence, the correct answer is option: 'D'.

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