Let f, g: R R be two functions defined by f(x) , = , * 20 c x , ,… - Vidyadip

MCQ

Let $f, g: R \to R$ be two functions defined by $f(x)\, = \,\left\{ {\begin{array}{*{20}{c}}
{x\,\sin \,\left( {\frac{1}{x}} \right),\,x\, \ne \,0\,\,\,\,\,\,\,\,\,\,}\\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x\, = 0\,\,\,\,\,\,\,\,\,}
\end{array}} \right.,$ and $g(x) =x\,f(x)$

Statement $I:$ $f$ is a continuous function at $x = 0.$
Statement $II:$ $g$ is a differentiable function at $x = 0.$

A
Both statement $I$ and $II$ are false.
✓
Both statement $I$ and $II$ are true.
C
Statement $I$ is true, statement $II$ is false.
D
Statement $I$ is false, statement $II$ is true.

✓

Answer

Correct option: B.

Both statement $I$ and $II$ are true.

b
$f\left( x \right) = \left\{ \begin{array}{l}
x\sin \left( {\frac{1}{x}} \right),\,\,\,x \ne 0\\
0\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0
\end{array} \right.$

and $g(x)=xf(x)$

For $f(x)$

$LHL = \mathop {\lim }\limits_{h \to {0^ - }} \left\{ { - h\sin \left( { - \frac{1}{h}} \right)} \right\}$

$ = 0 \times a$ finite quatity between $-1$ and $1$

$=0$

$RHL = \mathop {\lim }\limits_{h \to {0^ + }} h\sin \frac{1}{h} = 0$

Also, $f(0)=0$

Thus $LHL = RHL = f\left( 0 \right)$

$\therefore $ $f(x)$ is continuous at $x=0$

$g\left( x \right) = \left\{ \begin{array}{l}
{x^2}\sin \frac{1}{x},\,\,\,x \ne 0\\
0\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0
\end{array} \right.$

For $g(x)$

$LHL = \mathop {\lim }\limits_{h \to {0^ - }} \left\{ { - {h^2}\sin \left( {\frac{1}{h}} \right)} \right\}$

$ = {0^2} \times a$ a finite quantify between $-1$ and $1$ $=0$

$RHL = \mathop {\lim }\limits_{h \to {0^ + }} {h^2}\sin \left( {\frac{1}{h}} \right) = 0$

Also $g(0)=0$

$\therefore $ $g(x)$ is continuous at $x=0$

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