Question

Tìm GTNN:

\(x+2y-2\sqrt{xy}-3\sqrt{x}-4\sqrt{y}+1\)

Answer 1

\(x+y+\frac{9}{4}-2\sqrt{xy}-3\sqrt{x}+3\sqrt{y}+y-7\sqrt{y}+\frac{49}{4}-\frac{27}{2}\)

\(=\left(\sqrt{x}-\sqrt{y}-\frac{3}{2}\right)^2+\left(\sqrt{y}-\frac{7}{2}\right)^2-\frac{27}{2}\ge-\frac{27}{2}\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}\sqrt{x}-\sqrt{y}-\frac{3}{2}=0\\\sqrt{y}=\frac{7}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=25\\y=\frac{49}{4}\end{matrix}\right.\)