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NDA 2017

Exam: NDA
Year: 2017
Subject: Mathematics
Chapter: Functions

NDA 2017 · Mathematics · Functions

If x is any real number, then (x^2)/(1+x^4) belongs to which one of...

If

x

is any real number, then

\frac{x^2}{1+x^4}

belongs to which one of the following intervals?

Options

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

Correct answer

B. $\left(0, \frac{1}{2}\right]$

Step-by-step 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}

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