\(G=x^2+y^2+xy+x+y=\left[x^2+x\left(y+1\right)+\dfrac{1}{4}\left(y+1\right)^2\right]+\dfrac{3}{4}\left(y^2+\dfrac{2}{3}y+\dfrac{1}{9}\right)-\dfrac{1}{3}\)
\(=\left(x+\dfrac{1}{2}y+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\left(y+\dfrac{1}{3}\right)^2-\dfrac{1}{3}\ge-\dfrac{1}{3}\)
\(minG=-\dfrac{1}{3}\Leftrightarrow\) \(\left\{{}\begin{matrix}x=-\dfrac{1}{3}\\y=-\dfrac{1}{3}\end{matrix}\right.\)
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