Ta có: A = 2x2 + 4y2 - 4xy - 4x - 4y + 15
= (x2 - 4xy + 4y2) + 2(x - 2y) + 1 + (x2 - 6x + 9) + 5
= (x - 2y)2 + 2(x - 2y) + 1 + (x - 3)2 + 5
= (x - 2y + 1)2 + (x - 3)2 + 5 \(\ge\)5 \(\forall\)x; y
Daaus "=" xảy ra <=> \(\left\{{}\begin{matrix}x-2y+1=0\\x-3=0\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}y=\frac{x+1}{2}\\x=3\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}y=2\\x=3\end{matrix}\right.\)
Vậy MinA = 5 khi x = 3 và y = 2
\(=2\left(x^2-2xy+y^2-2x+2y+1\right)+2\left(y^2-4y+4\right)+5\)
\(=\left(y-x+1\right)^2+2\left(y-2\right)^2+5\ge5\)
Vậy MIN=5 khi \(\left\{{}\begin{matrix}y=2\\x=3\end{matrix}\right.\)
