$\displaystyle \begin{array}{{>{\displaystyle}l}} A=\frac{\sqrt{x} +1}{\sqrt{x} -1} \ B=\left(\frac{1}{\sqrt{x} -1} +\frac{\sqrt{x}}{x-1}\right) .\frac{x-\sqrt{x}}{2\sqrt{x} +1} \ ( x\geqslant 0;x\#1) \ \\ a) \ Với\ x=\frac{9}{4} \ \rightarrow \sqrt{x} =\frac{3}{2} \ \\ Do\ đó\ A=\frac{\frac{3}{2} +1}{\frac{3}{2} -1} =5\ \\ b) B=\frac{\sqrt{x} +1+\sqrt{x}}{\left(\sqrt{x} -1\right)\left(\sqrt{x} +1\right)} .\frac{\sqrt{x}\left(\sqrt{x} -1\right)}{2\sqrt{x} +1} =\frac{\sqrt{x}}{\sqrt{x} +1} \ \\ c) \ AB=\frac{\sqrt{x}}{\sqrt{x} +1} .\frac{\sqrt{x} +1}{\sqrt{x} -1} =\frac{\sqrt{x} -1+1}{\sqrt{x} -1} =1+\frac{1}{\sqrt{x} -1} \ \ \ \\ Có\ \sqrt{x} -1\geqslant -1\ với\ mọi\ x\ \rightarrow \frac{1}{\sqrt{x} -1} \leqslant -1\ với\ mọi\ x\ \\ \rightarrow 1+\frac{1}{\sqrt{x} -1} \leqslant 1-1=0\ \\ Vậy\ Max\ AB=0\ tại\ x=0\ \\ \\ \\ \\ \end{array}$
