# Consider the function

Question:

Consider the function $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ defined by

$f(x)=\left\{\begin{array}{c}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, x \neq 0 \\ 0, x=0\end{array}\right.$. Then $\mathrm{f}$ is

1. (1) monotonic on $(-\infty, 0) \cup(0, \infty)$

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

3. (3) monotonic on $(0, \infty)$ only

4. (4) monotonic on $(-\infty, 0)$ only

Correct Option: , 2

Solution:

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

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

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

$f^{\prime}(x)$ is an oscillating function which is non-monotonic in

$(-\infty, 0) \cup(0, \infty)$