Let f: R R be a function defined by f(x)= cc x^2 ( x^2 ) & if x 0… - Vidyadip

MCQ

Let $f: R \rightarrow R$ be a function defined by

$f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right) & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$

Then which of the following statements is $TRUE$?

A
$f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$.
B
$f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$.
C
The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite.
✓
$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.

✓

Answer

Correct option: D.

$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.

d
Option-$A$: $f ( x )= x ^2 \sin \frac{\pi}{ x ^2}$

$f ( x )=0 \Rightarrow \sin \frac{\pi}{ x ^2}=0 \Rightarrow \frac{\pi}{ x ^2}= n \pi, n \in N$

$x ^2=\frac{1}{ n } \Rightarrow x =\frac{1}{\sqrt{ n }}$

$\frac{1}{\sqrt{ n }} \geq \frac{1}{10^{10}} \Rightarrow 10^{10} \geq \sqrt{ n } \Rightarrow n \leq 10^{20}$, 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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