Đáp án:
`GTNNNN_P=38534/19<=>`
$ \begin{cases}y=-\dfrac{6}{9}\\x=\dfrac{12}{19}\\\end{cases}$
Giải thích các bước giải:
Sửa đề:
`P=5x^2+xy-6x+y^2+2030`
`4P=20x^2+4xy-24x+4y^2+8120`
`4P=x^2+4xy+4y^2+19x^2-24x+8120`
`4P=(x+2y)^2+19(x^2-24/19x)+8120`
`4P=(x+2y)^2+19(x^2-24/19x+144/361-144/361)+8120`
`4P=(x+2y)^2+19(x-12/19)^2+8120-144/19`
`4P=(x+2y)^2+19(x-12/19)^2+154136/19`
`(x+2y)^2>=0`
`19(x-12/19)^2>=0`
`=>(x+2y)^2+19(x-12/19)^2>=0`
`=>(x+2y)^2+19(x-12/19)^2+154136/19>=154136/19`
Hay `4P>=154136/19`
`=>P>=38534/19`
Dấu = xảy ra khi
$\begin{cases}x+2y=0\\x-\dfrac{12}{19}=0\\\end{cases}$
$ \Rightarrow \begin{cases}y=-\dfrac{1}{2}x=-\dfrac{6}{9}\\x=\dfrac{12}{19}\\\end{cases}$
Vậy `GTNNNN_P=38534/19<=>`
$ \Rightarrow \begin{cases}y=-\dfrac{6}{9}\\x=\dfrac{12}{19}\\\end{cases}$
