a, ĐKXĐ: \(x\ge0;x\ne9\)
b, rút gọn
A=\(\left(\dfrac{2\sqrt{x}}{\sqrt{x}+3}+\dfrac{\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x}{x-9}\right):\left(\dfrac{2\sqrt{x}}{\sqrt{x}-3}-1\right)\)
\(=\left[\dfrac{2\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}-\dfrac{3x+3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right]:\left(\dfrac{2\sqrt{x}-2}{\sqrt{x}-3}-\dfrac{\sqrt{x}+3}{\sqrt{x}-3}\right)\)
\(=\dfrac{2x-6\sqrt{x}+x+3\sqrt{x}+3x-3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}:\dfrac{\sqrt{x}+1}{x-3}\)
\(=\dfrac{-3\sqrt{x}-3}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}.\dfrac{\sqrt{x}-3}{x+1}\\ =\dfrac{-3\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}-\dfrac{\sqrt{x}-3}{\sqrt{x}+1}\\ =\dfrac{-3}{\sqrt{x}+3}\)
c,Cho \(A\le-\dfrac{1}{3}\)
\(< =>\dfrac{3}{\sqrt{x}+3}\le-\dfrac{1}{3}\\ < =>\dfrac{-3}{\sqrt{x}+3}+\dfrac{1}{3}\le0\\ < =>\dfrac{-9+\sqrt{x}+3}{3\left(\sqrt{x}+3\right)}\le0\\ < =>\dfrac{\sqrt{x}-6}{3\left(\sqrt{x}+3\right)}\le0\\ < =>\sqrt{x}-6\le0\\ < =>\sqrt{x}\le36\\ < =>0\le x\le36\)
Vậy để \(A\le-\dfrac{1}{3}\) thì \(0\le x\le36\)và\(x\ne9\)
d, \(A=\dfrac{-3}{\sqrt{x}+3}\)
Ta có: \(\sqrt{x}+3\ge3\\ =>\dfrac{1}{\sqrt{x}+3}\le\dfrac{1}{3}\\ =>\dfrac{-3}{\sqrt{x}+3}\ge\dfrac{-3}{3}\\ =-1\)
Vậy GTNN của A=-1
Xấu ''='' xảy ra khi \(\sqrt{x}=0\\ \Leftrightarrow x=0\)