\(P=\frac{\left(\sqrt{x}-1\right)\left(x-\sqrt{x}-2\right)}{\sqrt{x}+1}=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}{\sqrt{x}+1}\)

\(P=\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)=x-3\sqrt{x}+2\)

\(P=\left(\sqrt{x}-\frac{3}{2}\right)^2-\frac{1}{4}\ge-\frac{1}{4}\)

\(P_{Min}=-\frac{1}{4}\) khi \(\sqrt{x}=\frac{3}{2}\Leftrightarrow x=\frac{9}{4}\)

b/ \(Q=\frac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(-x+3\sqrt{x}-2\right)}=\frac{\sqrt{x}-1}{-\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}=\frac{1}{\left(\sqrt{x}+1\right)\left(2-\sqrt{x}\right)}\)

\(Q\ge\frac{1}{\frac{\left(\sqrt{x}+1+2-\sqrt{x}\right)^2}{4}}=\frac{4}{3^2}=\frac{4}{9}\)

\(Q_{min}=\frac{4}{9}\) khi \(\sqrt{x}+1=2-\sqrt{x}\Leftrightarrow x=\frac{1}{4}\)

c/ \(R=\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}-1}=\sqrt{x}+2+\frac{2}{\sqrt{x}-1}\)

Chắc là bạn ghi nhầm đề, với \(x< 1\) biểu thức này ko có min

Nó chỉ có min khi \(x>1\)

Khi đó: \(R=\sqrt{x}-1+\frac{2}{\sqrt{x}-1}+3\ge2\sqrt{\frac{2\left(\sqrt{x}-1\right)}{\sqrt{x}-1}}+3=3+2\sqrt{2}\)

\(R_{min}=3+2\sqrt{2}\) khi \(\sqrt{x}-1=\sqrt{2}\Leftrightarrow x=3+2\sqrt{2}\)

Chứng tỏ 0<Q<2 nha

\(P=\frac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-2\sqrt{x}-1+2\sqrt{x}+1=\frac{x^2-\sqrt{x}}{x+\sqrt{x}+1}\)

\(P+1=\frac{x^2+x+1}{x+\sqrt{x}+1}=\frac{x^2+2x+1-x}{x+\sqrt{x}+1}=\frac{\left(x+\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{x+\sqrt{x}+1}=x-\sqrt{x}+1\ge\frac{3}{4}\)

a/ \(P=\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}+1}{\sqrt{x}-3}-\frac{3-11\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)

     \(=\frac{2\sqrt{x}\left(\sqrt{x}-3\right)+\left(\sqrt{x}+1\right)\left(\sqrt{x}+3\right)-\left(3-11\sqrt{x}\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)

       \(=\frac{2x-6\sqrt{x}+x+4\sqrt{x}+3-3+11\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)

         \(=\frac{3x+9\sqrt{x}}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}=\frac{3\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}=\frac{3\sqrt{x}}{\sqrt{x}-3}\)

b/ \(P< 1\Rightarrow\frac{3\sqrt{x}}{\sqrt{x}-3}< 1\Rightarrow\frac{2\sqrt{x}+3}{\sqrt{x}-3}< 0\)

   Xét 2 trường hợp:

• \(\hept{\begin{cases}2\sqrt{x}+3>0\\\sqrt{x}-3< 0\end{cases}\Rightarrow\hept{\begin{cases}2\sqrt{x}>-3\\\sqrt{x}< 3\end{cases}\Rightarrow}\hept{\begin{cases}\sqrt{x}>-\frac{3}{2}\\\sqrt{x}< 3\end{cases}}\Rightarrow-\frac{3}{2}< \sqrt{x}< 3}\)

                                       \(\Rightarrow-\frac{9}{4}< x< 9\)

•  \(\hept{\begin{cases}2\sqrt{x}+3< 0\\\sqrt{x}>3\end{cases}\Rightarrow\hept{\begin{cases}\sqrt{x}< -\frac{3}{2}\\\sqrt{x}>3\end{cases}}}\) (vô lí)

                                                   Vậy -9/4 < x < 9