Bài 6:
\(F=-x^2+2xy-4y^2+2x+10y-3\)
\(=-x^2+2xy-y^2+2x-2y-3y^2+12y-3\)
\(=-\left(x-y\right)^2+2\left(x-y\right)-1-3y^2+12y-2\)
\(=-\left(x-y-1\right)^2-3y^2+12y-12+10\)
\(=-\left(x-y-1\right)^2-3\left(y^2-4y+4\right)+10\)
\(=-\left(x-y-1\right)^2-3\left(y-2\right)^2+10\le10\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}y-2=0\\ x-y-1=0\end{cases}\Rightarrow\begin{cases}y=2\\ x=y+1=2+1=3\end{cases}\)
Bài 8:
\(H=-x^2+xy-y^2-2x+4y+11\)
\(=\frac12\left(-2x^2+2xy-2y^2-4x+8y+22\right)\)
\(=\frac12\left(-x^2+2xy-y^2-x^2-4x-4-y^2+8y-16+50\right)\)
\(=\frac12\left\lbrack-\left(x-y\right)^2-\left(x+2\right)^2-\left(y-4\right)^2+50\right\rbrack=-\frac12\left(x-y\right)^2-\frac12\left(x+2\right)^2-\frac12\left(y-4\right)^2+25\le25\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}x-y=0\\ x+2=0\\ y-4=0\end{cases}\Rightarrow\begin{cases}x=y\\ x=-2\\ y=4\end{cases}\) (vô lý)
=>H không có giá trị lớn nhất
