Q63 / 150 · MHT CET 2023 (10 May Shift 2) · Mathematics

Let $\mathrm{S}=\left\{\mathrm{t} \in \mathrm{R} / \mathrm{f}(x)=|x-\pi|\left(\mathrm{e}^{|x|}-1\right) \sin |x|\right.$ is not differentiable at $\mathrm{t}\}$, then $\mathrm{S}$ is

  • Correct answer
    A
    $\phi$ (an empty set)
  • B
    $\{0\}$
  • C
    $\{\pi\}$
  • D
    $\{0, \pi\}$

Solution

Differentiability at $x=\pi$ : L.h.lim $=\lim _{h \rightarrow 0} \frac{|\pi-h-\pi|\left(e^{|\pi-h|}-1\right) \sin |\pi-h|-0}{-h}=0$ R.h.lim $=\lim _{h \rightarrow 0} \frac{|\pi+h-\pi|\left(e^{|\pi+h|}-1\right) \sin |\pi+h|-0}{h}=0$ Differentiability at $\boldsymbol{x}=\mathbf{0}$ : L.h.lim $=\lim _{\mathrm{h} \rightarrow 0} \frac{|-\mathrm{h}-\pi|\left(\mathrm{e}^{|-\mathrm{h}|}-1\right) \sin |-\mathrm{h}|-0}{-\mathrm{h}}=0$ R.h.lim $=\lim _{h \rightarrow 0} \frac{|h-\pi|\left(e^{|h|}-1\right) \sin |h|-0}{h}=0$ The function $\mathrm{f}(x)$ is differentiable at $x=0, \pi$. $\Rightarrow$ Set $\mathrm{S}$ is an empty set.

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

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