Let f : be a function defined by f ( x ) = { x 2 sin ( x 2 ) , if x 0 0 , if x = 0 Then which of the following statements is TRUE? [2024]

Question

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

$f (x)$ $=$ ${\begin{matrix} x^{2} \sin (\frac{π}{x^{2}}), & if x \neq 0 \\ 0, & if x = 0 \end{matrix}$

Then which of the following statements is TRUE? [2024]

Smart Achievers · Accepted Answer

(4)

Given, $f (x)$ $=$ ${\begin{matrix} x^{2} \sin (\frac{π}{x^{2}}), & if x \neq 0 \\ 0, & if x = 0 \end{matrix}$

$f (x) = 0 \Rightarrow \sin (\frac{π}{x^{2}}) = 0$

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

$(a) If x \in [\frac{1}{10^{10}}, \infty)$

$\frac{1}{\sqrt{n}} \in [\frac{1}{10^{10}}, \infty)$

$\sqrt{n} \in (0, 10^{10}]$

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

$Finite values of n are possible, so has finite solution.$

$(b) If x \in [\frac{1}{π}, \infty) \Rightarrow \frac{1}{\sqrt{n}} \in [\frac{1}{π}, \infty)$

$\sqrt{n} \in (0, π] \Rightarrow n \in (0, π^{2}]$ $\Rightarrow n = 1, 2, 3, \dots, 9$

$(c) If x \in (0, \frac{1}{10^{10}}) \Rightarrow \sqrt{n} \in (10^{10}, \infty)$

$n is infinite$

$(d) If x \in (\frac{1}{π^{2}}, \frac{1}{π}) \Rightarrow \sqrt{n} \in (π, π^{2})$

$n \in (π^{2}, π^{4})$ $\Rightarrow n \in (9.8, 97.2, \dots)$

$More than 25 solutions$

Want Us to Email you About Special Offers & Updates?