\(A=12xy+6x-13x^2-9y^2+5\)
\(\Leftrightarrow A=-4x^2+12xy-9y^2-9x^2+6x-1+6\)
\(\Leftrightarrow A=-\left(4x^2-12xy+9y^2\right)-\left(9x^2-6x+1\right)+6\)
\(\Leftrightarrow A=-\left[\left(2x\right)^2-2.2x.3y+\left(3y\right)^2\right]-\left[\left(3x\right)^2-2.3x.1+1^2\right]+6\)
\(\Leftrightarrow A=-\left(2x-3y\right)^2-\left(3x-1\right)^2+6\)
Vậy GTLN của \(A=6\) khi \(\left\{{}\begin{matrix}2x-3y=0\\3x-1=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2.\dfrac{1}{3}-3y=0\\x=\dfrac{1}{3}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2}{9}\\x=\dfrac{1}{3}\end{matrix}\right.\)
\(A=12xy+6x-13x^2-9y^2+5\)
\(\Leftrightarrow A=-4x^2+12xy-9y^2-9x^2+6x-1+6\)
\(\Leftrightarrow A=-\left(4x^2-12xy+9y^2\right)-\left(9x^2-6x+1\right)+6\)
\(\Leftrightarrow A=-\left[\left(2x\right)^2-2.2x.3y+\left(3y\right)^2\right]- \left[\left(3x\right)^2-2.3x.1+1^2\right]+6\)
\(\Leftrightarrow A=-\left(2x-3y\right)^2-\left(3x-1\right)^2+6\)
Vậy GTLN của \(A=6\) khi \(\left\{{}\begin{matrix}2x-3y=0\\3x-1=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}2.\dfrac{1}{3}-3y=0\\x=\dfrac{1} {3}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{2}{9}\\x=\dfrac{1} {3}\end{matrix}\right.\)
