\(F=-x^2-2y^2+2xy-y+1\)
\(-F=x^2+2y^2-2xy+y-1\)
\(-F=\left(x^2-2xy+y^2\right)+\left(y^2+y+\frac{1}{4}\right)-\frac{5}{4}\)
\(-F=\left(x-y\right)^2+\left(y+\frac{1}{2}\right)^2-\frac{5}{4}\)
Mà \(\left(x-y\right)^2\ge0\forall x;y\)
\(\left(y+\frac{1}{2}\right)^2\ge0\forall y\)
\(\Rightarrow-F\ge-\frac{5}{4}\)
\(\Leftrightarrow F\le\frac{5}{4}\)
Dấu "=" xảy ra khi :
\(\hept{\begin{cases}x-y=0\\y+\frac{1}{2}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-\frac{1}{2}\\y=-\frac{1}{2}\end{cases}}\)
Vậy \(F_{Max}=\frac{5}{4}\Leftrightarrow x=y=-\frac{1}{2}\)
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