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Let \(f:\mathrm{ℝ}\to \mathrm{ℝ}\) be a function defined by \(f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^…

Q1

Let \(f:\mathrm{ℝ}\to \mathrm{ℝ}\) 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?

[JEE Advanced 2024]

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.

d

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

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