\(ĐKXĐ:x>\dfrac{1}{4}\)
Áp dụng BĐT Cauchy cho các số dương , ta có :
\(\dfrac{x}{\sqrt{4x-1}}+\dfrac{\sqrt{4x-1}}{x}\ge2\sqrt{\dfrac{x}{\sqrt{4x-1}}.\dfrac{\sqrt{4x-1}}{x}}=2\)
\("="\Leftrightarrow\dfrac{x}{\sqrt{4x-1}}=\dfrac{\sqrt{4x-1}}{x}\Leftrightarrow x^2=4x-1\)
\(\Leftrightarrow x^2-4x+4-3=0\Leftrightarrow\left(x-2\right)^2-3=0\)
\(\Leftrightarrow\left(x-2+\sqrt{3}\right)\left(x-2-\sqrt{3}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2-\sqrt{3}\left(KTM\right)\\x=2+\sqrt{3}\left(TM\right)\end{matrix}\right.\)
KL.....
\(\dfrac{x}{\sqrt{4x-1}}+\dfrac{\sqrt{4x-1}}{x}=2\Leftrightarrow\dfrac{x^2+4x-1}{x\sqrt{4x-1}}=2\)
