\(A=\left(x-y\right)^2+\left(x+1\right)^2+\left(y-5\right)^2+2001\)
\(=2x^2+2y^2-2xy+2x-10y+2027\)
\(=\left(2x^2-2x\left(y-1\right)+\dfrac{\left(y-1\right)^2}{2}\right)-\dfrac{\left(y-1\right)^2}{2}+2y^2-10y+2027\)
\(=2\left(x^2-x\left(y-1\right)+\dfrac{\left(y-1\right)^2}{4}\right)+\dfrac{3}{2}y^2-9y+\dfrac{4053}{2}\)
\(=2\left(x-\dfrac{y-1}{2}\right)^2+\dfrac{3}{2}\left(y-3\right)^2+2013\ge2013\)
Dấu '' = '' xảy ra khi \(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=3\end{matrix}\right.\)
Vậy: Min A = 2013 tại \(x=1;y=3.\)
\(A=\left(x-y\right)^2+\left(x+1\right)^2+\left(y-5\right)^2+2001\ge2001\)Vậy Min A = 2001 khi \(\left[{}\begin{matrix}\left(x-y\right)^2=0\\\left(x+1\right)^2=0\\\left(y-5\right)^2=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=y\\x=-1\\y=5\end{matrix}\right.\)
\(C=4x^2+3y^2-2xy-10x-14y+30\)
\(=4x^2-2x\left(y+5\right)+\dfrac{\left(y+5\right)^2}{4}-\dfrac{\left(y+5\right)^2}{4}+3y^2-14y+30\)
\(=\left(2x-\dfrac{y+5}{2}\right)^2+\dfrac{11}{4}y^2-\dfrac{33}{2}y+\dfrac{95}{4}\)
\(=\left(2x-\dfrac{y+5}{2}\right)^2+\dfrac{11}{4}\left(y-3\right)^2-1\ge-1\)
Dấu '' = '' xảy ra khi \(\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=3\end{matrix}\right.\)
Vậy: Min C = -1 tại \(x=2;y=3.\)