Đáp án:
$min_S= -12 \Leftrightarrow \left\{\begin{array}{l} x=-1 \\ y=-1\end{array} \right..$
Giải thích các bước giải:
$x+y=-2 \Rightarrow y=-x-2\\ S=2(x^3+y^3)-15xy+7\\ =2(x^3+(-x-2)^3)-15x(-x-2)+7\\ =2(x^3-(x+2)^3)+15x(x+2)+7\\ =2(x^3-(x^3 + 6 x^2 + 12 x + 8))+15x(x+2)+7\\ =2(- 6 x^2 - 12 x - 8)+15x^2+30x+7\\ =- 12 x^2 - 24 x - 16+15x^2+30x+7\\ =3 x^2 + 6 x - 9\\ =3( x^2 +2x-3)\\ =3( x^2 +2x+1-4)\\ =3( x^2 +2x+1)-12\\ =3(x+1)^2-12 \ge -12 \ \forall \ x$
Dấu "=" xảy ra $\Leftrightarrow \left\{\begin{array}{l} x+1 =0 \\ y=-x-2\end{array} \right. \Leftrightarrow \left\{\begin{array}{l} x=-1 \\ y=-1\end{array} \right..$
