Let f : R → R be a function defined by f(x) = { {x^2 sin(π/x^2), if x ≠ 0} {0, if x = 0}

Question

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) 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.

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

Answer 1

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

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

\(f(x) =0 \Rightarrow \sin \left(\frac{\pi}{x^2}\right) = 0\)

\(\Rightarrow \frac{\pi}{x^{2}}=n \pi \)

\( \Rightarrow x^{2}=\frac{1}{n}\)

\(\Rightarrow x=\frac{1}{\sqrt{n}}\)

\(\text {If } x \in\left[\frac{1}{10^{10}}, \infty\right) \)

\(\frac{1}{\sqrt{n}} \in\left[\frac{1}{10^{10}}, \infty\right) \)

\(\sqrt{n} \in\left(0,10^{10}\right] \)

\(n \in\left(0,\left(10^{10}\right)^{2}\right]\)

\(\text {Finite values of } n \)

\(\text {If } x \in\left[\frac{1}{\pi}, \infty\right) \)

\( \frac{1}{\sqrt{n}} \in\left[\frac{1}{\pi}, \infty\right)\)

\( \sqrt{n} \in(0, \pi]\)

\(n \in\left(0, \pi^{2}\right] \)

\(n=1,2,3 \ldots 9 \)

\(\text {If } x \in\left(0, \frac{1}{10^{10}}\right)\)

\(\sqrt{n} \in\left(10^{10}, \infty\right)\)

\(n \text { infinite }\)

\(\text {If } x \in\left(\frac{1}{\pi^{2}}, \frac{1}{\pi}\right) \)

\( \sqrt{n} \in\left(\pi, \pi^{2}\right) \)

\(n \in\left(\pi^{2}, \pi^{4}\right) \)

\( n \in(9.8,97.2 \ldots)\)

More than 25 solutions