Câu 1:
`P=4x^2+y^2-2x-10y+2xy-3`
`=(4x^2-2x+2xy)+y^2-10y-3`
`=4(x^2-1/2x+1/2xy)+y^2-10y-3`
`=4[x^2+2x(1/4y-1/4)+(1/4y-1/4)^2]-4(1/4y-1/4)^2+y^2-10y-3`
`=4(x+1/4y-1/4)^2-4(1/16y^2-1/8y+1/16)+y^2-10y-3`
`=4(x+1/4y-1/4)^2-1/4y^2+1/2y-1/4+y^2-10y-3`
`=4(x+1/4y-1/4)^2+3/4y^2-19/2y-13/4`
`=4(x+1/4y-1/4)^2+3/4y^2-19/2y+361/12-100/3`
`=4(x+1/4y-1/4)^2+3/4(y-38/3y+361/9)-100/3`
`=4(x+1/4y-1/4)^2+3/4(y-19/3)^2-100/3\ge -100/3`
Đẳng thức xảy ra `<=>x=-4/3;\ y=19/3`
Vậy `min P=-100/3` đạt được khi `x=-4/3;\ y=19/3`
Câu 2:
`G=x^2+y^2+xy-3x-3y`
`=(x^2+xy-3x)+y^2-3y`
`=[x^2+2x(1/2y-3/2)+(1/2y-3/2)^2]-(1/2y-3/2)^2+y^2-3y`
`=(x+1/2y-3/2)^2-1/4y^2+3/2y-9/4+y^2-3y`
`=(x+1/2y-3/2)^2+3/4y^2-3/2y-9/4`
`=(x+1/2y-3/2)^2+3/4y^2-3/2y+3/4-3`
`=(x+1/2y-3/2)^2+3/4(y^2-2y+1)-3`
`=(x+1/2y-3/2)^2+3/4(y-1)^2-3\ge -3`
Đẳng thức xảy ra `<=>x=y=1`
Vậy `min G=-3` đạt được khi `x=y=1`
