Question

Tìm GTNN của A=2x² +9y² -6xy -6x-12y +2014

Answer 1

\(A=x^2+\left(3y\right)^2+2^2-6xy+4x-12y+x^2-10x+25+1985\)

\(A=\left(x-3y+2\right)^2+\left(x-5\right)^2+1985\ge1985\)

\(\Rightarrow A_{min}=1985\) khi \(\left\{{}\begin{matrix}x=5\\y=\frac{7}{3}\end{matrix}\right.\)

Answer 2

A = \(2x^2+9y^2-6xy-6x-12y+2014\)

\(=\left(x^2-6xy+9y^2\right)+4\left(x-3y\right)+4+\left(x^2-10x+25\right)+100+1885\)

\(=\left(x-3y+2\right)^2+\left(x-5\right)^2+1885\ge1885\)

Vậy GTNN của A = 1885 khi

\(\left\{{}\begin{matrix}x-3y+2=0\\x-5=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x=5\\x-3y+2=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x=5\\y=\frac{7}{3}\end{matrix}\right.\)