Question

Consider the function
$f\left(x\right)=\{\begin{array}{cc}x\sin \frac{\pi }{x},& x>0\\ x,& x=0\end{array}$
The number of points in $(0,1),$ where the $f^{\prime }\left(x\right)$ vanishes, is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is D infinite

$f\left(x\right)$ vanishes at points where
$\sin \frac{\pi }{x}=0$,
i.e., $\frac{\pi }{x}=k\pi ,k=1,2,3,4,\cdots$
Hence $x=\frac{1}{k}$
Also $f^{\prime }\left(x\right)=\sin \frac{\pi }{x}-\frac{\pi }{x}\cos \frac{\pi }{x}$, if $x\ne 0$
Since the function has a derivative at any interior point of the interval $(0,1)$, also continuous in $[0,1]$ and $f\left(0\right)=f\left(1\right)$
Hence, Rolle's theorem is applicable to any one of the interval $[\frac{1}{2},1],[\frac{1}{3},\frac{1}{2}],\cdots ,[\frac{1}{k+1},\frac{1}{k}]$
Hence, there exists at least one $c$ in each of these intervals where $f^{\prime }\left(c\right)=0\Rightarrow$ infinite points.