Ta có: \(S\geq\frac{4}{3} \Leftrightarrow \frac{x}{y+1}+\frac{y}{x+1}\geq \frac{4}{3}\Leftrightarrow \frac{x^2+y^2+x+y}{xy+x+y+1}\geq\frac{4}{3}\Leftrightarrow3(8+x+y)\geq4(xy+x+y+1)\Leftrightarrow x+y+4xy\leq20\)(luôn đúng do \(x+y\leq\sqrt{2(x^2+y^2}=4;4xy\leq2(x^2+y^2)=16\)).
Vậy...
\(x^2+y^2\ge\frac{1}{2}\left(x+y\right)^2\Rightarrow x+y\le\sqrt{2\left(x^2+y^2\right)}=4=x^2+y^2\)
\(S=\frac{x^2}{xy+x}+\frac{y^2}{xy+y}\ge\frac{\left(x+y\right)^2}{2xy+x+y}\ge\frac{\left(x+y\right)^2}{2xy+x^2+y^2}=\frac{\left(x+y\right)^2}{\left(x+y\right)^2}=1\)
\(S_{min}=1\) khi \(x=y=2\)
. Em cảm ơn
