`M = x^2 - 5x + y^2 + xy - 4y + 2014`
`= y^2 + y(x - 4) + x^2 - 5x + 2014`
`= y^2 + 2.y.(x - 4)/2 + ((x - 4)/2)^2 - ((x - 4)/2)^2 + x^2 - 5x + 2014`
`= (y + (x - 4)/2)^2 + (4x^2 - 20x - x^2 + 8x - 16)/4 + 2014`
`= (y + (x - 4)/2)^2 + (3x^2 - 12x + 12 - 28)/4 + 2014`
`= (y + (x - 4)/2)^2 + 3(x - 2)^2/4 + 2007`
Vì` {(y + (x - 4)/2)^2 ≥ 0 <=> {(y + (x - 4)/2)^2 ≥ 0`
`{(x - 2)^2 ≥ 0 {3(x - 2)^2/4 ≥ 0`
`=> (y + (x - 4)/2)^2 + 3(x - 2)^2/4 + 2007 ≥ 2007`
Dấu `"="` xảy ra `<=> (y + (x - 4)/2)^2 = (x - 2)^2 = 0 <=> x = 2; y = 1`