Question

Tìm GTNN của:

A=x²+y²+xy-x+y+1

Biết xy=11 và x²y+xy²+x+y=2010. Tính x²+y^2.

Answer 1

Sửa đề

\(2A=2x^2+2y^2+2xy-2x+2y+2\)

\(=\left(x^2+2xy+y^2\right)+\left(x^2-2x+1\right)+\left(y^2+2y+1\right)\)

\(=\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2\ge0\)

\(\Rightarrow A_{min}=0\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\)

Answer 2

\(x^2y+xy^2+x+y=xy\left(x+y\right)+\left(x+y\right)=\left(x+y\right)\left(xy+1\right)=12\left(x+y\right)=2010\)

\(\Rightarrow x+y=\dfrac{2010}{12}\)

\(\Rightarrow x^2+y^2=\left(x+y\right)^2-2xy=\left(\dfrac{2010}{12}\right)^2-2\cdot11=\dfrac{112137}{4}\)