Question

tìm GTNN của C = \(x^2-2xy-4x+2y^2-8y+20\)

Answer 1

\(C=x^2-2xy-4x+2y^2-8y+20\)

\(=\left(x^2-2xy+y^2\right)-4\left(x-y\right)+4+y^2-12y+36-20\)

\(=\left(x-y\right)^2-4\left(x-y\right)+4+\left(y-6\right)^2-20\)

\(=\left(x-y-2\right)^2+\left(y-6\right)^2-20\ge-20\forall x;y\)

Dấu " = " xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x-y-2=0\\y-6=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-y=2\\y=6\end{matrix}\right.\)

\(\Leftrightarrow x=8;y=6\)

Vậy Min C là : \(-20\Leftrightarrow x=8;y=6\)