\(\dfrac{\sqrt{a}}{\dfrac{\sqrt{a}}{\sqrt{b}}-1}\ge\dfrac{\sqrt{b}}{1-\dfrac{\sqrt{b}}{\sqrt{a}}}\Leftrightarrow\dfrac{\sqrt{a}-\sqrt{b}-\sqrt{a}+\sqrt{b}}{\left(\dfrac{\sqrt{a}}{\sqrt{b}}-1\right)\left(1-\dfrac{\sqrt{b}}{\sqrt{a}}\right)}\ge0\)
\(\Leftrightarrow\left(\dfrac{\sqrt{a}}{\sqrt{b}}-1\right)\left(1-\dfrac{\sqrt{b}}{\sqrt{a}}\right)>0\Leftrightarrow\dfrac{\sqrt{a}}{\sqrt{b}}-1-1+\dfrac{\sqrt{b}}{\sqrt{a}}\ge0\)
\(\Leftrightarrow\dfrac{\sqrt{a}}{\sqrt{b}}+\dfrac{\sqrt{b}}{\sqrt{a}}-2\ge0\Leftrightarrow\dfrac{\sqrt{a}}{\sqrt{b}}+\dfrac{\sqrt{b}}{\sqrt{a}}\ge2\) (Đúng theo AM-GM)
Vậy bđt cần cm đúng
\("="\Leftrightarrow a=b\)
