\(A=2x^2+2xy+y^2-2x+2y+2\)
\(\Rightarrow2A=4x^2+4xy+2y^2-4x+4y+4\)
\(=\left(4x^2+4xy+y^2\right)-2\left(2x+y\right).1+1+y^2+6y+9-6\)
\(=\left(2x+y\right)^2-2\left(2x+y\right)+1+\left(y+3\right)^2-6\)
\(=\left(2x+y-1\right)^2+\left(y+3\right)^2-6\)
vì \(\left(2x+y-1\right)^2\ge0\forall x,y;\left(y+3\right)^2\ge0\forall y\)nên
\(2A=\left(2x+y-1\right)+\left(y+3\right)-6\ge-6\forall x,y\)
hay \(2A\ge-6\Rightarrow A\ge-3\Rightarrow minA=-3\Leftrightarrow\hept{\begin{cases}2x+y-1=0\\y+3=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=2\\y=-3\end{cases}}}\)
