If f(x) , = , * 20 c x e^ - , ( 1 | ,x ,| , + ,1 x ) , & x 0 0 , … - Vidyadip

MCQ

If $f(x)\, = \,\left\{ {\begin{array}{*{20}{c}}{x{e^{ - \,\left( {\frac{1}{{|\,x\,|}}\, + \,\frac{1}{x}} \right)}},}&{x \ne 0}\\{0\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x = 0}\end{array}} \right.$ , then $f(x)\,$ is

A
Continuous as well as differentiable for all $x$
✓
Continuous for all $x$ but not differentiable at $x = 0$
C
Neither differentiable nor continuous at $x = 0$
D
Discontinuous every where

✓

Answer

Correct option: B.

Continuous for all $x$ but not differentiable at $x = 0$

b
(b) $f(0) = 0$ and $f(x) = x{e^{ - \left( {\frac{1}{{|x|}} + \frac{1}{x}} \right)}}$

$R.H.L. =$ $\mathop {\lim }\limits_{h \to 0} (0 + h){e^{ - 2/h}} = \mathop {\lim }\limits_{h \to 0} \frac{h}{{{e^{2/h}}}} = 0$

$L.H.L. =$ $\mathop {\lim }\limits_{h \to 0} (0 - h){e^{ - \left( {\frac{1}{h}\, - \,\frac{1}{h}} \right)}} = 0$; $\therefore$ $f(x)$ is continuous.

$Rf'\,(x) = \mathop {\lim }\limits_{h \to 0} \frac{{(0 + h){e^{ - \left( {\frac{1}{h} + \frac{1}{h}} \right)}} - h{e^{ - \left( {\frac{1}{h} + \frac{1}{h}} \right)}}}}{h} = 0$

$Lf'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{(0 - h){e^{ - \left( {\frac{1}{h} - \frac{1}{h}} \right)}} - h{e^{ - \left( {\frac{1}{h} + \frac{1}{h}} \right)}}}}{{ - h}} = 1$

==> $Lf'(x) \ne Rf'(x)$. $f(x)$ is not differentiable at $x = 0.$

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