Question

Tìm x, y để các biểu thức sau đạt GTNN

A = x² + 2y² - 2xy - 4y + 10

Answer 1

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

\(=\left(x-y\right)^2+\left(y-2\right)^2+6\ge6\)

\(minA=6\Leftrightarrow x=y=2\)

Answer 2

\(A=x^2+2y^2-2xy-4y+10\\ =x^2-2xy+y^2+y^2-4y+4+6\\ =\left(x-y\right)^2+\left(y-2\right)^2+6\ge6\forall x,y\in R\)

\(Min_A=6\Leftrightarrow\left\{{}\begin{matrix}x-y=0\\y-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y\\y=2\end{matrix}\right.\Leftrightarrow x=y=2\)