\(C=2x^2+9y^2-6xy-2x+2018\)
\(=\left(x^2-6xy+9y^2\right)+\left(x^2-2x+1\right)+2017\)
\(=\left(x-3y\right)^2+\left(x-1\right)^2+2017\)
Nhận xét :
\(\left\{{}\begin{matrix}\left(x-3y\right)^2\ge0\\\left(x-1\right)^2\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left(x-3y\right)^2+\left(x-1\right)^2\ge0\)
\(\Leftrightarrow\left(x-3y\right)^2+\left(x-1\right)^2+2017\ge2017\)
\(\Leftrightarrow C\ge2017\)
Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\left(x-3y\right)^2=0\\\left(x-1\right)^2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{1}{3}\end{matrix}\right.\)
Vậy \(C_{Min}=2017\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{1}{3}\end{matrix}\right.\)
\(D=x^2-2xy+6y^2-12x+2y+45\)
\(=\left(x^2-2xy+y^2\right)-\left(12x+12y\right)-10y+5y^2+45\)
\(=\left(x-y\right)^2-12\left(x-y\right)+36+\left(5y^2-10y+5\right)+4\)
\(=\left(x-y-6\right)^2+5\left(y-1\right)^2+4\)
Nhận xét :
\(\left\{{}\begin{matrix}\left(x-y-6\right)^2\ge0\\5\left(y-1\right)^2\ge0\end{matrix}\right.\)
\(\Leftrightarrow\left(x-y-6\right)+5\left(y-1\right)^2+4\ge4\)
\(\Leftrightarrow D\ge4\)
Dấu "=" xảy ra khi \(\Leftrightarrow\left\{{}\begin{matrix}x=7\\y=1\end{matrix}\right.\)
Vậy \(D_{Min}=4\Leftrightarrow\left\{{}\begin{matrix}x=7\\y=1\end{matrix}\right.\)
