The function f ( x ) = e | x | is, Continuity and Differentiability Mock Test Questions JEE Main

Question

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

Smart Achievers · Accepted Answer

(1)

Ans. Continuous everywhere but not differentiable at $x = 0$

Explanation:

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

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

$f (0) = e^{0} = 1$

$RHL = \lim_{x \to 0^{+}} e^{x} = e^{0} = 1$

$LHL = \lim_{x \to 0^{-}} e^{- x} = e^{- 0} = 1$

If $x = 0$ ,

$f (0) = LHL = RHL$

$∴$ $f (x)$ is continuous at $x = 0$

$LHD = \lim_{x \to 0^{-}} f' (x) = \lim_{x \to 0^{-}} (- e^{- x}) = - e^{- 0} = - 1$

$RHD = \lim_{x \to 0^{+}} e^{x} = e^{0} = 1$

$LHD \neq RHD$

$∴$ $f (x)$ is not differentiable at $x = 0$

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