Consider the function f (x) = l x , x , , ,for , ,x , > 0 0 , , ,… - Vidyadip

MCQ

Consider the function $f (x) =\left\{ \begin{array}{l} x\,\sin \frac{\pi }{x}\,\,\,for\,\,x\, > 0\\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,\,x\, = \,0 \end{array} \right.$ then the number of points in $(0, 1)$ where the derivative $ f '(x)$ vanishes , is

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

✓

Answer

$f (x)$ vanishes at points where $\sin \frac{\pi}{x}= 0$
i.e. $\frac{\pi}{x}= k\pi , k = 1, 2, 3, 4, .....$
hence $x = \frac{1}{k} $.
Also $f ' (x) = \sin\frac{\pi}{x}- \frac{\pi}{x} \cos\frac{\pi}{x}$ 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
$f (0) = f (1)$
hence Rolle's theorem is applicable to any one of the interval $\left[ {\frac{1}{2}\,,\,1} \right]$,
$\left[ {\frac{1}{3}\,,\,\frac{1}{2}\,} \right], …. \left[ {\frac{1}{{k + 1}}\,,\,\frac{1}{k}\,} \right]$
hence some $c$ in each of these interval where $f\ ' (c) = 0$
$ \Rightarrow $infinite points
$ \Rightarrow(D) $

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