Question

Tìm Min P = x - 2\(\sqrt{xy}\) + 3y - 2\(\sqrt{x}\) + 1

Answer 1

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

\(\Leftrightarrow3P=3x-6\sqrt{xy}+9y-6\sqrt{x}+3\)

\(=\left(x-6\sqrt{xy}+9y\right)+\left(2x-\dfrac{2.\sqrt{2}.3.\sqrt{x}}{\sqrt{2}}+\dfrac{9}{2}\right)-\dfrac{3}{2}\)

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

\(\Rightarrow P\ge-\dfrac{1}{2}\)

Vậy GTNN là \(P=-\dfrac{1}{2}\) đạt được khi \(\left\{{}\begin{matrix}x=\dfrac{9}{4}\\y=\dfrac{1}{4}\end{matrix}\right.\)