`P=x^2-5x-1`
`P=x^2-2.x. 5/2+ 25/4 - 25/4 - 1`
`P=[x^2-2x. 5/2 + (5/2)^2] - 29/4`
`P=(x-5/2)^2 - 29/4`
Có: `(x-5/2)^2\ge0⇒ (x-5/2)^2 - 29/4\ge0 + (-29/4) = -29/4.`
Dấu ''='' xảy ra khi `x-5/2= 0 ⇔ x=5/2.`
Vậy $Min_P=$ `-29/4 ⇔ x=5/2.`
`M= 2x^2 + 9y^2 - 6xy - 6x +2004 - 12y`
`M=x^2+x^2+9y^2-6xy+4x -10x + 25 + 4 + 1975-12y`
`M=(x^2-6xy+9y^2)+(4x-12y)+4+(x^2-10x+25)+1975`
`M=(x-3y)^2+4(x-3y)+4 + (x-5)^2+1975`
`M=(x-3y+2)^2 + (x-5)^2+1975`
Có `(x-3y+2)^2\ge0, (x-5)^2\ge0 ⇒ (x-3y+2)^2 + (x-5)^2+1975\ge1975`
Dấu ''='' xảy ra khi `x-3y+2=0, x-5=0 ⇔ x=5 ⇒ y= 7/3.`
Vậy $Min_M=$ `1975 ⇔ x=5, y=7/3.`
