\(G=x^2-2xy+2y^2+2x-10y+17\\ \\ =x^2-2xy+y^2+y^2+2x-2y-8y+1+16\\ \\ =\left(x^2+y^2+1-2xy+2x-2y\right)+\left(y^2-8y+16\right)\\ \\ =\left(x-y+1\right)^2+\left(y-4\right)^2\)
Do \(\left(x-y+1\right)^2\ge0\forall x;y\)
\(\left(y-4\right)^2\ge0\forall y\)
\(\Rightarrow G=\left(x-y+1\right)^2+\left(y-4\right)^2\ge0\forall x;y\)
Dấu \("="\) xảy ra khi: \(\left\{{}\begin{matrix}\left(x-y+1\right)^2=0\\\left(y-4\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y+1=0\\y-4=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=y-1\\y=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=4\end{matrix}\right.\)
Vậy \(G_{\left(Min\right)}=0\) khi \(\left\{{}\begin{matrix}x=3\\y=4\end{matrix}\right.\)
\(H=x^2+2xy+y^2-2x-2y\\ =x^2+2xy+y^2-2x-2y+1-1\\ =\left(x^2+y^2+1+2xy-2x-2y\right)-1\\ \\ =\left(x+y-1\right)^2-1\)
Do \(\left(x+y-1\right)^2\ge0\forall x;y\)
\(\Rightarrow H=\left(x+y-1\right)^2-1\ge-1\forall x;y\)
Dấu \("="\) xảy ra khi:
\(\left(x+y-1\right)^2=0\\ \Leftrightarrow x+y-1=0\\ \Leftrightarrow x+y=1\)
Vậy \(H_{\left(Min\right)}=-1\) khi \(x+y=1\)
