Đặt \(A = x^2 + y^2 - xy - 3x - 3y + 2021\)
\(\Rightarrow2A=2x^2+2y^2-2xy-6x-6y+4042\)
\(=\left(x^2-2xy+y^2\right)+\left(x^2-6x+9\right)+\left(y^2-6y+9\right)+4024\)
\(=\left(x-y\right)^2+\left(x-3\right)^2+\left(y-3\right)^2+4024\)
Mà \(\left(x-y\right)^2,\left(x-3\right)^2,\left(y-3\right)^2\ge0\)
\(\Rightarrow2A\ge4024\Leftrightarrow A\ge2012\)
Vậy GTNN của A là 2012 khi x = y = 3
Đặt A= x2 + y2 - xy -3x -3y + 2021
=> 2A= 2x2 +2y2 -2xy - 6x -6y + 4024
=> 2A= (x2 -2xy +y2) +( x2 - 6x +9) +(y2 -6y +9) + 4006
=> 2A= (x-y)2 +(x -3)2 +(y- 3)2 +4006
vì (x-y)2 + (x -3)2 + (y -3)2 \(\ge\) 0 với mọi x,y
=> 2A\(\ge\)4006 => A\(\ge\)2003
Dấu "=" xảy ra <=> \(\left\{{}\begin{matrix}x-y=0\Rightarrow x=y\\x-3=0\Rightarrow x=3\\y-3=0\Rightarrow y=3\end{matrix}\right.\) Vậy GTNN cửa A= 2003 khi x=y=3
