Consider the function f : R R defined by f(x)= cc (2- (1 x ) )|x|… - Vidyadip

MCQ

Consider the function $f : R \rightarrow R$ defined by $f(x)=\left\{\begin{array}{cc}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, x \neq 0 \\ 0 & , x=0\end{array} .\right.$ Then $f$ is

A
monotonic on $(-\infty, 0) \cup(0, \infty)$
✓
not monotonic on $(-\infty, 0)$ and $(0, \infty)$
C
monotonic on $(0, \infty)$ only
D
monotonic on $(-\infty, 0)$ only

✓

Answer

Correct option: B.

not monotonic on $(-\infty, 0)$ and $(0, \infty)$

b
$f(x)=\left\{\begin{array}{cl}-x\left(2-\sin \left(\frac{1}{x}\right)\right) & x<0 \\ 0 & x=0 \\ x\left(2-\sin \left(\frac{1}{x}\right)\right) & \end{array}\right.$

$f ^{\prime}( x )=\left\{\begin{array}{ll}-\left(2-\sin \frac{1}{ x }\right)- x \left(-\cos \frac{1}{ x } \cdot\left(-\frac{1}{ x ^{2}}\right)\right) & x <0 \\ \left(2-\sin \frac{1}{ x }\right)+ x \left(-\cos \frac{1}{ x }\left(-\frac{1}{ x ^{2}}\right)\right) & x >0\end{array}\right.$

$f^{\prime}(x)=\left\{\begin{array}{l}-2+\sin \frac{1}{x}-\frac{1}{x} \cos \frac{1}{x} x<0 \\ 2-\sin \frac{1}{x}+\frac{1}{x} \cos \frac{1}{x} x>0\end{array}\right.$

$f^{\prime}(x)$ is an oscillating function which is non-monotonic in $(-\infty, 0) \cup(0, \infty)$.

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