\(P=\left(\dfrac{\sqrt{x}}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}-1}-\dfrac{2\sqrt{x}}{x-1}\right).\dfrac{\sqrt{x}+1}{\sqrt{x}+2}\left(Đk:x\ge0;x\ne1\right)\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)+\sqrt{x}+1-2\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}.\dfrac{\sqrt{x}+1}{\sqrt{x}+2}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}.\dfrac{\sqrt{x}+1}{\sqrt{x}+2}=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}\)
c) Khi \(x=\dfrac{1}{9}\Rightarrow\sqrt{x}=\dfrac{1}{3}\)
\(P=\dfrac{\dfrac{1}{3}-1}{\dfrac{1}{3}+2}=-\dfrac{2}{7}\)
d) \(P< -\dfrac{1}{3}\)
\(\dfrac{\sqrt{x}-1}{\sqrt{x}+2}< -\dfrac{1}{3}\)
\(\dfrac{\sqrt{x}-1}{\sqrt{x}+2}+\dfrac{1}{3}< 0\)
\(\dfrac{3\left(\sqrt{x}-1\right)+\sqrt{x}+2}{3\left(\sqrt{x}+2\right)}< 0\)
\(\dfrac{4\sqrt{x}-1}{3\left(\sqrt{x}+2\right)}< 0\)
Vì \(\sqrt{x}+2>0\Rightarrow4\sqrt{x}-1< 0\Rightarrow\sqrt{x}< \dfrac{1}{4}\)
\(\Rightarrow0\le x< \dfrac{1}{16}\)
e) \(P=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}=1-\dfrac{3}{\sqrt{x}+2}\)
Vì \(\sqrt{x}+2\ge2\Rightarrow\dfrac{3}{\sqrt{x}+2}\le\dfrac{3}{2}\)
\(\Rightarrow P=1-\dfrac{3}{\sqrt{x}+2}\ge1-\dfrac{3}{2}=-\dfrac{1}{2}\)
\(MinP=-\dfrac{1}{2}\Leftrightarrow x=0\)
