$f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right) & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$
द्वारा परिभाषित है। तब निम्नलिखित कथनों में से कौन सा सत्य है?
$f ( x )=0 \Rightarrow \sin \frac{\pi}{ x ^2}=0 \Rightarrow \frac{\pi}{ x ^2}= n \pi, n \in N$
$x ^2=\frac{1}{ n } \Rightarrow x =\frac{1}{\sqrt{ n }}$
$\frac{1}{\sqrt{ n }} \geq \frac{1}{10^{10}} \Rightarrow 10^{10} \geq \sqrt{ n } \Rightarrow n \leq 10^{20}$, finite number of solutions
Option-$B$ : $x =\frac{1}{\sqrt{ n }} \Rightarrow \frac{1}{\sqrt{ n }}>\frac{1}{\pi} \Rightarrow \pi>\sqrt{ n } \Rightarrow n <\pi^2$, Number of solutions is 9
Option-$C$: $x =\frac{1}{\sqrt{ n }}, \frac{1}{\sqrt{ n }}<\frac{1}{10^{10}} \Rightarrow \sqrt{ n }>10^{10} \Rightarrow n >10^{20}$. Infinite number of solutions
Option-$D$ : $\frac{1}{\pi^2}<\frac{1}{\sqrt{ n }}<\frac{1}{\pi} \Rightarrow \sqrt{ n } \in\left(\pi, \pi^2\right) \Rightarrow n \in\left(\pi^2, \pi^4\right)$, Definitely more than $25$ solutions
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