\(M=x^2+5y^2+2xy-2y+2017\)
\(\Leftrightarrow M=x^2+y^2+4y^2+2xy-2y+\dfrac{1}{4}+\dfrac{8067}{4}\)
\(\Leftrightarrow M=\left(x^2+2xy+y^2\right)+\left(4y^2-2y+\dfrac{1}{4}\right)+\dfrac{8067}{4}\)
\(\Leftrightarrow M=\left(x^2+2xy+y^2\right)+\left[\left(2y\right)^2-2.2y.\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2\right]+\dfrac{8067}{4}\)
\(\Leftrightarrow M=\left(x+y\right)^2+\left(2y-\dfrac{1}{2}\right)^2+\dfrac{8067}{4}\)
Vậy GTNN của \(M=\dfrac{8067}{4}\) khi \(\left\{{}\begin{matrix}x+y=0\\2y-\dfrac{1}{2}=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{4}=0\\y=\dfrac{1}{4}\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{1}{4}\\y=\dfrac{1}{4}\end{matrix}\right.\)
