\(M=x^2+2y^2+2xy-2x-3y+1\)
=> \(M=x^2+2x\left(y-1\right)+\left(y-1\right)^2-\left(y-1\right)^2+2y^2-3y+1\)
=> \(M=\left(x+y-1\right)^2-y^2+2y-1+2y^2-3y+1\)
=> \(M=\left(x+y-1\right)^2+y^2-y\)
=> \(M=\left(x+y-1\right)^2+y^2-2y\frac{1}{2}+\frac{1}{4}-\frac{1}{4}\)
=> \(M=\left(x+y-1\right)^2+\left(y-\frac{1}{2}\right)^2-\frac{1}{4}\)
Có \(\left(x+y-1\right)^2\ge0\)với mọi x, y
\(\left(y-\frac{1}{2}\right)^2\ge0\)với mọi y
=> \(M=\left(x+y-1\right)^2+\left(y-\frac{1}{2}\right)^2-\frac{1}{4}\ge\frac{-1}{4}\)với mọi x, y
Dấu "=" xảy ra <=> \(\hept{\begin{cases}x+y-1=0\\y-\frac{1}{2}=0\end{cases}}\)
<=> \(\hept{\begin{cases}x=\frac{1}{2}\\y=\frac{1}{2}\end{cases}}\)
KL: Mmin = \(\frac{-1}{4}\)<=> \(x=y=\frac{1}{2}\)
