P = x2 + 2y2 - 2xy + 8x + 8y + 2017
= x2 + y2 + 42 - 2xy - 8y + 8x + y2 + 16y + 64 + 1937
= (x - y + 4)2 + (y + 8)2 + 1937 \(\ge\) 1937
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}x-y+4=0\\y+8=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x=-12\\y=-8\end{matrix}\right.\)
\(P=x^2+2y^2-2xy+8x+8y+2017\)
\(=\left(x^2-2xy+8x\right)+2y^2+8y+2017\)
\(=\left[x^2-2x\left(y-8\right)+\left(y-8\right)^2\right]+2y^2+8y+2017-y^2+16y-64\)\(=\left(x-y+8\right)^2+y^2+24y+1953\)
\(=\left(x-y+8\right)^2+\left(y^2+24y+144\right)+1809\)
\(=\left(x-y+8\right)^2+\left(y+12\right)^2+1809\ge1809\forall x\)Vậy Min P = 1809 khi \(\left\{{}\begin{matrix}x-y+8=0\\y+12=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x+20=0\\y=-12\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=-20\\y=-12\end{matrix}\right.\)
