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If x is anyreal number, then $\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}$ belongs to which one of the following intervals?
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Verified Answer
The correct answer is:
$\left(0, \frac{1}{2}\right]$
$\mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}} \Rightarrow \mathrm{y} \geq 0 .$
Also, $\mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}=\frac{1}{\mathrm{x}^{2}+\frac{1}{\mathrm{x}^{2}}}$
$\Rightarrow \mathrm{y} \leq \frac{1}{2}$
$\therefore \frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}$ belongs to $\left(0, \frac{1}{2}\right]$
Also, $\mathrm{y}=\frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}=\frac{1}{\mathrm{x}^{2}+\frac{1}{\mathrm{x}^{2}}}$
$\Rightarrow \mathrm{y} \leq \frac{1}{2}$
$\therefore \frac{\mathrm{x}^{2}}{1+\mathrm{x}^{4}}$ belongs to $\left(0, \frac{1}{2}\right]$
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