Question

Consider the function $f:R\to R$ defined by $f(x)=\{\begin{array}{l}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|,x\ne 0\\ 0,x=0\end{array}$ Then $f$ is:

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

Answer 1

Answer: (B)

$f(x)=\{\begin{array}{ll}-x\left(2-\sin \left(\frac{1}{x}\right)\right)& x<0\\ 0& x=0\\ x\left(2-\sin \left(\frac{1}{x}\right)\right)& \end{array}$
$f^{\prime }(x)=\{\begin{array}{ll}-\left(2-\sin \frac{1}{x}\right)-x\left(-\cos \frac{1}{x}\cdot \left(-\frac{1}{x^2}\right)\right)& x<0\\ \left(2-\sin \frac{1}{x}\right)+x\left(-\cos \frac{1}{x}\left(-\frac{1}{x^2}\right)\right)& x>0\end{array}$
$f^{\prime }(x)=\{\begin{array}{l}-2+\sin \frac{1}{x}-\frac{1}{x}\cos \frac{1}{x}x<0\\ 2-\sin \frac{1}{x}+\frac{1}{x}\cos \frac{1}{x}x>0\end{array}$
$f^{\prime }(x)$ is an oscillating function which is non-monotonic in $(-\infty ,0)\cup (0,\infty )$.