Inequalities are Amazing!

$\frac {1}{1 + x^2} \geq \frac {y - x}{y + x}$

$(y + x) \geq (y - x)(1 + x^2)$

$y + x \geq y - x +x^2y - x^3$

$2x \geq x^2y - x^3$

$xy \leq 2 + x^2$

$y \leq \frac {2}{x} + x$

Define a function $f(x)= \frac {2}{x} + x$

$f'(x)=-2x^{-2} + 1$

$-2x^{-2} + 1 = 0$

$x= \pm \sqrt{2}$

$f''(x)=4x^{-3}$

$f''(\sqrt{2})= \frac {4}{(\sqrt{2})^3} >0$

Therefore the minimum value of x when x>0 is $\sqrt{2}$ .

So $y \leq \frac {2}{\sqrt{2}} + \sqrt{2}$

$y = 2 \sqrt{2} = 0.282...$

Just wanted to practice my latex haha