Q234 / 270 · NDA 2021 (Phase 1) · Mathematics

If $f(x)=e^{|x|}$, then which one of the following is correct?

  • A
    $f^{\prime}(0)=1$
  • B
    $f^{\prime}(0)=-1$
  • C
    $f^{\prime}(0)=0$
  • Correct answer
    D
    $f^{\prime}(0)$ does not exist

Solution

$\begin{aligned} & f(x)=e^{|x|} \\ & f^{\prime}(x)=e^x, x \gt 0\end{aligned}$ $\begin{gathered}=-e^{|x|}, x \lt 0 \\ \lim _{x \rightarrow 0^{+}} f^{\prime}(x) \neq \lim _{x \rightarrow 0^{-}} f^{\prime}(x)\end{gathered}$ $\therefore f^{\prime}(x)$ does not exist at $x=0$

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

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