Ta có:
A=-2x^2-10y^2+4xy +4x+4y+2013
=-(2x^2+10^2-4xy-4x-4y-2013)
=-[(2x^2+2y^2-4xy)-(4x-4y)+2-2015+8y^2-8y]
=-[2(x-y)^2-4(x-y)+2+(8y^2-8y+2)-2017]
=-[2(x-y-1)^2+8(y-1/4)^2]+2017
vì 2(x-y-1)^2\(\ge\)0với mọi x,y
8(y-1/4)^2\(\ge\)0với mọi y
=>-[2(x-y-1)^2+8(y-1/4)^2]\(\le\)0với mọi x,y
=>A=-[2(x-y-1)^2+8(y-1/4)^2]+2017\(\le\)2017với mọi x,y
dấu "=" xảy ra khi\(\Leftrightarrow\left\{{}\begin{matrix}y-\dfrac{1}{4}=0\\x-y-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{1}{4}\\x-\dfrac{5}{4}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{1}{4}\\x=\dfrac{5}{4}\end{matrix}\right.\)
Vậy GTLN của A là 2017 khi y=1/4;x=5/4
