Max and Min

Let $x,y>0$. Show that \[\max\{\min\{x,y,\frac{1}{x}+\frac{1}{y}\}\}=\min\{\max\{x,y,\frac{1}{x}+\frac{1}{y}\}\}=\sqrt{2}\]

Proof:

\[\left(\max\{x,y,\frac{1}{x}+\frac{1}{y}\}\right)^2\geqslant\frac{(x+y)(\frac{1}{x}+\frac{1}{y})}{2}\geqslant2\]

Surpose that $y\geqslant x$, we have \[\left(\min\{x,y,\frac{1}{x}+\frac{1}{y}\}\right)^2\leqslant x\left(\frac{1}{x}+\frac{1}{y}\right)=1+\frac{x}{y}\leqslant2\]

Then it shows that $\max=\sqrt{2}$