\(A=-x^2-4xy-5y^2-6y+1672\)
\(=-x^2-4xy-4y^2-y^2-6y-9+1681\)
\(=-\left(x^2+4xy+4y^2\right)-\left(y^2+6y+9\right)+1681\)
\(=-\left(x+2y\right)^2-\left(y+3\right)^2+1681\)
Dễ thấy: \(\left\{{}\begin{matrix}\left(x+2y\right)^2\ge0\forall x,y\\\left(y+3\right)^2\ge0\forall y\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}-\left(x+2y\right)^2\le0\forall x,y\\-\left(y+3\right)^2\le0\forall y\end{matrix}\right.\)
\(\Rightarrow-\left(x+2y\right)^2-\left(y+3\right)^2\le0\forall x,y\)
\(\Rightarrow A=-\left(x+2y\right)^2-\left(y+3\right)^2+1681\le1681\forall x,y\)
Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}-\left(x+2y\right)^2=0\\-\left(y+3\right)^2=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x+2y=0\\y+3=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x=-2y\\y=-3\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=6\\y=-3\end{matrix}\right.\)
