Q228 / 270 · NDA 2017 (Phase 2) · Mathematics

If $x$ is any real number, then $\frac{x^2}{1+x^4}$ belongs to which one of the following intervals?

  • A
    $(0,1)$
  • Correct answer
    B
    $\left(0, \frac{1}{2}\right]$
  • C
    $\left(0, \frac{1}{2}\right)$
  • D
    $[0,1]$

Solution

$y=\frac{x^2}{1+x^4} \Rightarrow y \geq 0$ Also, $y=\frac{x^2}{1+x^4}=\frac{1}{x^2+\frac{1}{x^2}}$ $\begin{aligned} & \Rightarrow \mathrm{y} \leq \frac{1}{2} . \\ & \therefore \frac{\mathrm{x}^2}{1+\mathrm{x}^4} \text { belongs to }\left(0, \frac{1}{2}\right] .\end{aligned}$

Hence, the correct answer is option: 'B'.

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