P = x2 + 2y2 + 2xy – 6x – 8y + 2028
P = (x2 + y2 + 2xy) – 6(x + y) + 9 + y2 – 2y + 1 + 2018
P = (x + y – 3)2 + (y – 1)2 + 2018 \(\ge\) 2018
=> Giá trị nhỏ nhất của P = 2018 khi x = 2; y = 1
P=x2+2y2+2xy-6x-8y+2028
=x2+2xy+y2+y2-8y+x2-6x-x2+2028
=(x2+2xy+y2)+(y2-8y+16)+(x2-6x+9)-x2+2028-16-9
=(x-y)2+(y-4)2+(x-3)2-x2+2003\(\ge2003\)
Vì \(\left\{{}\begin{matrix}\left(x-y\right)^2\ge0\\\left(y-4\right)^2\ge0\\\left(x-3\right)^2\ge0\\x^2\ge0\end{matrix}\right.\) nên:
Để P=2003 thì :
\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-y\right)^2=0\\\left(x-3\right)^2=0\\\left(y-4\right)^2=0\\x^2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\x-3=0\\y-4=0\\x=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=y\\x=3\\y=4\\x=0\end{matrix}\right.\)
Vậy min P=2003\(\Leftrightarrow\left(x=y\right)\in\left\{0;4;3\right\}\)
P = x2 + 2y2 + 2xy – 6x – 8y + 2028
P = (x2 + y2 + 2xy) – 6(x + y) + 9 + y2 – 2y + 1 + 2018
P = (x + y – 3)2 + (y – 1)2 + 2018 ≥≥ 2018
=> Giá trị nhỏ nhất của P = 2018 khi x = 2; y = 1
