Ta có \(A=\dfrac{2x+3y}{2x+y+2}\Leftrightarrow2Ax+Ay+2A-2x-3y=0\Leftrightarrow2A=2x-2Ax+3y-Ay\Leftrightarrow2A=2x\left(1-A\right)+y\left(3-A\right)\Leftrightarrow\left(2A\right)^2=\left[2x\left(1-A\right)+y\left(3-A\right)\right]^2\left(1\right)\)Áp dụng bđt bunhiacopski ta có \(\left[2x\left(1-A\right)+y\left(3-A\right)\right]^2\le\left(4x^2+y^2\right)\left[\left(1-A\right)^2+\left(3-A\right)^2\right]\Leftrightarrow\left(2A\right)^2\le1.\left(1-2A+A^2+9-6A+A^2\right)\Leftrightarrow4A^2\le2A^2-8A+10\Leftrightarrow2A^2+8A-10\le0\Leftrightarrow A^2+4A-5\le0\Leftrightarrow A^2-A+5A-5\le0\Leftrightarrow A\left(A-1\right)+5\left(A-1\right)\le0\Leftrightarrow\left(A-1\right)\left(A+5\right)\le0\Leftrightarrow\)\(\left[{}\begin{matrix}\left\{{}\begin{matrix}A-1\le0\\A+5\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}A-1\ge0\\A+5\le0\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow\)\(\left[{}\begin{matrix}\left\{{}\begin{matrix}A\le1\\A\ge-5\end{matrix}\right.\\\left\{{}\begin{matrix}A\ge1\\A\le-5\end{matrix}\right.\left(ktm\right)\end{matrix}\right.\)
Vậy \(-5\le A\le1\)
Vậy GTNN của A là -5
GTLN của A là 1
