The function f ( x ) = e | x | is :

Question

The function $f (x) = e^{| x |}$ is :

Smart Achievers · Accepted Answer

(4)

Ans. $f' (0)$ does not exist.

Explanation:

$f (x) = e^{| x |}$ $\Rightarrow$ $f (x) =$ ${\begin{matrix} e^{x}, & x \geq 0 \\ e^{- x}, & x < 0 \end{matrix}$

$RHD = \lim_{h \to 0} \frac{e^{h} - f (0)}{h}$

$\lim_{h \to 0} \frac{e^{h} - 1}{h} = 1$

$LHD = \lim_{h \to 0} \frac{e^{- h} - f (0)}{- h}$

$= \lim_{h \to 0} \frac{e^{- h} - 1}{- h} = - 1$

Since, $RHD \neq LHD$

that implies $f$ is not differentiable at $x = 0$ .

$f' (0)$ does not exist.

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