\[\begin{array}{l}
S = \left( {{x^3} - 1} \right)\left( {{y^3} - 1} \right),\,\,\,x,\,\,y \ge 0,\,\,x + y = 4.\\
S = {x^3}{y^3} - \left( {{x^3} + {y^3}} \right) + 1 = {\left( {xy} \right)^3} - \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) + 1\\
= {\left( {xy} \right)^3} - 4\left[ {{{\left( {x + y} \right)}^2} - 3xy} \right] + 1\\
= {\left( {xy} \right)^3} - 4\left( {16 - 3xy} \right) + 1\\
= {\left( {xy} \right)^3} + 12xy - 63.\\
Dat\,\,\,xy = t\,\,\,\,\left( {0 \le t \le 4} \right)\\
\Rightarrow S = {t^3} + 12t - 63\\
\Rightarrow S' = 3{t^2} + 12 > 0\,\,\forall t\\
\Rightarrow hs\,\,DB\\
\Rightarrow \mathop {Max}\limits_{\left[ {0;\,\,4} \right]} y = y\left( 4 \right) = 49\\
Dau\,\, = \,\,xay\,\,ra \Leftrightarrow xy = 4\\
\left\{ \begin{array}{l}
xy = 4\\
x + y = 4
\end{array} \right. \Leftrightarrow x = y = 2.
\end{array}\]
