\[\begin{array}{l}
x,\,\,y > 0;\,\,\,x + y = 2\\
Ta\,\,\,co:\\
xy\left( {{x^2} + {y^2}} \right) = xy\left( {{x^2} + 2xy + {y^2} - 2xy} \right)\\
= xy\left[ {{{\left( {x + y} \right)}^2} - 2xy} \right] = xy\left( {4 - 2xy} \right)\\
= 4xy - 2{\left( {xy} \right)^2}\\
= - 2\left[ {{{\left( {xy} \right)}^2} - 2xy + 1} \right] + 2\\
= - 2{\left( {xy - 1} \right)^2} + 2 \le 2\,\,\,\forall xy.\\
\Rightarrow xy\left( {{x^2} + {y^2}} \right) \le 2\,\,\forall x,\,\,y > 0.
\end{array}\]
