Let f(x)=x x |, x R. Then, - Vidyadip

MCQ

Let $f(x)=x \mid \sin x |, x \in R$. Then,

A
$f$ is differentiable for all $x$, except at $x=n \pi, n=1,2,3$,
✓
$f$ is differentiable for all $x$, except at $x=n \pi, n=\pm 1, \pm 2, \pm 3, \ldots$
C
$f$ is differentiable for all $x$, except at $x=n \pi, n=0,1,2,3$
D
$f$ is differentiable for all $x$, except at $x=n \pi, n=0, \pm 1, \pm 2, \pm 3, \ldots$

✓

Answer

Correct option: B.

$f$ is differentiable for all $x$, except at $x=n \pi, n=\pm 1, \pm 2, \pm 3, \ldots$

b
(b)

We have, $f(x)=x|\sin x|, x \in R$

$f(x)=\left\{\begin{array}{ll} x \sin x, & x \in(2 n \pi,(2 n+1) \pi) \\ -x \sin x, & x \in((2 n+1) \pi, 2 n \pi) \end{array}\right.$

$f^{\prime}(n \pi)=\lim _{x \rightarrow n \pi} \frac{f(x)-f(n \pi)}{x-n \pi}$

$=\lim _{x \rightarrow n \pi} \frac{x \sin x \mid}{x-n \pi}$

Clearly, $f(x)$ is differentiable for all $x$ except $x=n \pi, n=\pm 1, \pm 2, \pm 3, \ldots$

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