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 ==> $infinite points $ ==>(D) $
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${l_1} = \left( {3 + t} \right)\hat i + \left( { - 1 + 2t} \right)\hat j + \left( {4 + 2t} \right)\hat k,\, - \infty < t < \infty $
${l_2} = \left( {3 + 2s} \right)\hat i + \left( {3 + 2s} \right)\hat j + \left( {2 + s} \right)\hat k,\, - \infty < s < \infty $
Statement $1$ : Line $l$ and $l_2$ are coplaner lines
Statement $2$ : Line $l$ and $l_2$ are intersecting lines
