Sửa đề: \(A=\left(\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-3}-\frac{3x+3}{x-9}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)
a) ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne9\end{matrix}\right.\)
b) Ta có: \(A=\left(\frac{2\sqrt{x}}{\sqrt{x}+3}+\frac{\sqrt{x}}{\sqrt{x}-3}-\frac{3x+3}{x-9}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-1\right)\)
\(=\left(\frac{2\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\frac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\frac{3x+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right):\left(\frac{2\sqrt{x}-2}{\sqrt{x}-3}-\frac{\sqrt{x}-3}{\sqrt{x}-3}\right)\)
\(=\left(\frac{2x-6\sqrt{x}+x+3\sqrt{x}-3x-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\right):\left(\frac{2\sqrt{x}-2-\sqrt{x}+3}{\sqrt{x}-3}\right)\)
\(=\frac{-3\sqrt{x}-3}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}:\frac{\sqrt{x}+1}{\sqrt{x}-3}\)
\(=\frac{-3\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\cdot\frac{\sqrt{x}-3}{\sqrt{x}+1}\)
\(=\frac{-3}{\sqrt{x}+3}\)
c) Để \(A\le-\frac{1}{3}\) thì \(A+\frac{1}{3}\le0\)
\(\Leftrightarrow\frac{-3}{\sqrt{x}+3}+\frac{1}{3}\le0\)
\(\Leftrightarrow\frac{-9}{3\left(\sqrt{x}+3\right)}+\frac{\sqrt{x}+3}{3\left(\sqrt{x}+3\right)}\le0\)
\(\Leftrightarrow\frac{\sqrt{x}-6}{3\left(\sqrt{x}+3\right)}\le0\)
⇔\(\sqrt{x}-6\) và \(3\left(\sqrt{x}+3\right)\) cùng dấu
mà \(3\left(\sqrt{x}+3\right)>0\forall x\) thỏa mãn ĐKXĐ
nên \(\sqrt{x}-6< 0\)
\(\Leftrightarrow\sqrt{x}< 6\)
\(\Leftrightarrow\left|x\right|< 36\)
\(\Leftrightarrow-36< x< 36\)
Kết hợp ĐKXĐ, ta được: \(\left\{{}\begin{matrix}0\le x< 36\\x\ne9\end{matrix}\right.\)
Vậy: Để \(A\le-\frac{1}{3}\) thì \(\left\{{}\begin{matrix}0\le x< 36\\x\ne9\end{matrix}\right.\)
d) Ta có: \(\sqrt{x}+3\ge3\forall x\) thỏa mãn ĐKXĐ
⇔\(\frac{3}{\sqrt{x}+3}\le\frac{3}{3}=1\forall x\) thỏa mãn ĐKXĐ
\(\Leftrightarrow\frac{-3}{\sqrt{x}+3}\ge-1\forall x\) thỏa mãn ĐKXĐ
Dấu '=' xảy ra khi x=0
Vậy: Giá trị nhỏ nhất của A là -1 khi x=0
