Đáp án:
Max=36
Giải thích các bước giải:
\(\begin{array}{l}
R = - \left( {7{x^2} + 4{y^2} + 8xy - 18x - 9} \right)\\
= - \left( {4{x^2} + 8xy + 4{y^2} + 3{x^2} - 2.x\sqrt 3 .\dfrac{9}{{\sqrt 3 }} + \dfrac{{81}}{3} - 36} \right)\\
= - {\left( {2x + 2y} \right)^2} - {\left( {x\sqrt 3 - \dfrac{9}{{\sqrt 3 }}} \right)^2} + 36\\
Do:{\left( {2x + 2y} \right)^2} \ge 0\forall x;y\\
{\left( {x\sqrt 3 - \dfrac{9}{{\sqrt 3 }}} \right)^2} \ge 0\forall x;y\\
\to - {\left( {2x + 2y} \right)^2} - {\left( {x\sqrt 3 - \dfrac{9}{{\sqrt 3 }}} \right)^2} \le 0\\
\to - {\left( {2x + 2y} \right)^2} - {\left( {x\sqrt 3 - \dfrac{9}{{\sqrt 3 }}} \right)^2} + 36 \le 36\\
\to Max = 36\\
\Leftrightarrow \left\{ \begin{array}{l}
2x + 2y = 0\\
x\sqrt 3 - \dfrac{9}{{\sqrt 3 }} = 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
x = 3\\
y = - 3
\end{array} \right.
\end{array}\)
