Animated Solution for Mathematics - Trigonometry: Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\{\begin{array}{ll}{x}^2\sin \left(\frac{\pi }{x^2}\right)& ;\text{if }x\ne 0\\ 0& ;\text{if }x=0\end{array}$ Then which of the following statements is TRUE?

The function under investigation is defined as:

This function exhibits rapid oscillations as $x$ approaches the origin. To analyze its behavior, we must determine the points where the function vanishes.

To find the roots of $f(x)=0$ for $x>0$, we solve the equation:

Since $x^2 eq 0$ in this domain, the condition simplifies to $\sin \left(\frac{\pi }{x^2}\right)=0$. Recalling the general solution for the sine function, we set the argument to an integer multiple of $\pi$:

Canceling $\pi$ and rearranging for $x$, we obtain the roots:

We evaluate the interval $\left(\frac{1}{{\pi }^2},\frac{1}{\pi }\right)$ to determine the number of roots contained within it. We set up the following inequality:

Taking the reciprocal of all parts reverses the inequality signs:

Squaring the terms yields the range for the integer $n$:

Given the approximations ${\pi }^2\approx 9.87$ and ${\pi }^4\approx 97.4$, we seek integers $n$ such that $9.87<n<97.4$. This implies $n\in \{10,11,\dots ,97\}$.

The total number of roots is calculated as $97-10+1=88$. Since $88>25$, the statement is true.

For Option A, the interval $\left(\frac{1}{10^{10}},\infty \right)$ implies $\frac{1}{\sqrt{n}}>\frac{1}{10^{10}}$, or $n<10^{20}$. Because this set of integers is bounded, there are not infinitely many solutions.

For Option B, the interval $\left[\frac{1}{\pi },\infty \right)$ implies $\frac{1}{\sqrt{n}}\ge \frac{1}{\pi }$, leading to $n\le {\pi }^2\approx 9.86$. There are exactly 9 solutions, contradicting the claim that there are no solutions.

Finally, for Option C, the interval $\left(0,\frac{1}{10^{10}}\right)$ requires $\frac{1}{\sqrt{n}}<\frac{1}{10^{10}}$, or $n>10^{20}$. Since there are infinitely many such integers, the set of solutions is infinite, rendering the claim of a finite set false.