Đáp án:
Max=15
Giải thích các bước giải:
\(\begin{array}{l}
A = 5 + 2xy + 14y - {x^2} - 5{y^2} - 2x\\
= - ({x^2} + {y^2} + 1 - 2xy + 2x - 2y) - (4{y^2} - 12y + 9) + 5 + 1 + 9\\
= - {(x - y + 1)^2} - {(2y - 3)^2} + 15\\
Do:\left\{ \begin{array}{l}
{(x - y + 1)^2} \ge 0\forall x,y \in R\\
{(2y - 3)^2} \ge 0\forall x,y \in R
\end{array} \right.\\
\to \left\{ \begin{array}{l}
- {(x - y + 1)^2} \le 0\forall x,y \in R\\
- {(2y - 3)^2} \le 0\forall x,y \in R
\end{array} \right.\\
\to - {(x - y + 1)^2} - {(2y - 3)^2} \le 0\\
\to - {(x - y + 1)^2} - {(2y - 3)^2} + 15 \le 15\\
\to Max = 15\\
\Leftrightarrow \left\{ \begin{array}{l}
x - y + 1 = 0\\
2y - 3 = 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
y = \dfrac{3}{2}\\
x = \dfrac{5}{2}
\end{array} \right.
\end{array}\)
