\(B=-x^2-y^2-xy+2x+3y\)
\(=-x^2-x\left(y-2\right)-y^2+3y\)
\(=-\left(x^2\right)-2.x.\dfrac{y-2}{2}-\dfrac{\left(y-2\right)^2}{4}+\dfrac{y^2-4y+4}{4}-y^2+3y\)
\(=-\left(x+\dfrac{y-2}{2}\right)^2+\dfrac{-3y^2+8y+4}{4}\)
\(=-\left(x+\dfrac{y-2}{2}\right)^2+\dfrac{-3\left(y^2-2.y.\dfrac{4}{3}+\dfrac{16}{9}\right)+\dfrac{16}{3}+4}{4}\)
\(=-\left(x+\dfrac{y-2}{2}\right)^2-\dfrac{3\left(y-\dfrac{4}{3}\right)^2}{4}+\dfrac{\dfrac{28}{3}}{4}\)
\(=-\left(x+\dfrac{y-2}{2}\right)^2-\dfrac{3\left(y-\dfrac{4}{3}\right)^2}{4}+\dfrac{7}{3}\)
Vì \(\left\{{}\begin{matrix}-\left(x+\dfrac{y-2}{2}\right)^2\le0,\forall x,y\\-\dfrac{3\left(y-\dfrac{4}{3}\right)^2}{4}\le0,\forall y\end{matrix}\right.\)
\(\Rightarrow B\le\dfrac{7}{3}\)
\(MaxB=\dfrac{7}{3}\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{y-2}{2}\\y=\dfrac{4}{3}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}\\y=\dfrac{4}{3}\end{matrix}\right.\)