Question

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

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

Answer 1

Answer: (D)

infinite

The function vanishes at point where

$\sin \left(\frac{\pi }{x}\right)=0\Rightarrow \frac{\pi }{x}=k\pi \Rightarrow x=\frac{1}{k},k=1,2,3,...$

Since the function has derivative at any interior point of the interval $[0,1]$,

the Rolle's theorem is valid to anyone of the interval

$[\frac{1}{2},1],[\frac{1}{3},\frac{1}{2}],...[\frac{1}{k+1},\frac{1}{k}],...$

Consequently there exists $c_k\in (\frac{1}{k+1},\frac{1}{k})$

So that $f^{\prime }\left(c_k\right)=0.$

So the derivative vanishes at infinitely many points.