A = 2x2 + y2 - 2xy - 2x - 2y + 2015
= (x2 - 2xy + y2) + (x2 - 2x + 1) + (y2 - 2y + 1) + 2013
= (x - y)2 + (x - 1)2 + (y - 1)2 + 2013
Ta có: (x - y)2 \(\ge0\forall x;y\)
Dấu ''='' xảy ra khi (x - y)2 = 0 ⇔ x = y
(x - 1)2 \(\ge0\forall x\)
Dấu ''='' xảy ra khi (x - 1)2 = 0 ⇔ x = 1
(y - 1)2 \(\ge0\forall y\)
Dấu ''='' xảy ra khi (y - 1)2 = 0 ⇔ y = 1
Do đó: (x - y)2 + (x - 1)2 + (y - 1)2 + 2013 \(\ge2013\)
Hay A \(\ge2013\)
Dấu ''='' xảy ra khi x = y = 1
Vậy Min A = 2013 tại x = y = 1
\(A=2x^2+y^2-2xy-2x-2y+2015\\ =y^2-2y\left(x+1\right)+2x^2-2x+2015\\ =y^2-2y\left(x+1\right)+\left(x^2+2x+1\right)+\left(x^2-4x+4\right)+2010\\ =y^2-2y\left(x+1\right)+\left(x+1\right)^2+\left(x-2\right)^2+2010\\ =\left(y-x-1\right)^2+\left(x-2\right)^2+2010\ge2010\)
Min A = 2010 khi x=2 ; y=3
