`A=x/(1-x)+4/x` `(0<x<1)`
`A=x/(1-x)+(4/x-4)+4`
`A=x/(1-x)+(4-4x)/x+4`
`A=x/(1-x)+(4(1-x))/x+4`
Áp dụng BĐT Cosi:
`x/(1-x)+(4(1-x))/x>=2\sqrt{x/(1-x).(4(1-x))/x}`
`=>x/(1-x)+(4(1-x))/x>=2\sqrt4=4`
`=>x/(1-x)+(4(1-x))/x+4>=4+4=8`
Hay: `A>=8`
Dấu "=" xảy ra khi:
`x/(1-x)=(4(1-x))/x`
`=>x^2=4(1-x)^2`
`<=>`$\left[ \begin{array}{l}x=2(1-x)\\x=2(x-1)\end{array} \right.$
`<=>`$\left[ \begin{array}{l}x=2-2x\\x=2x-2\end{array} \right.$
`<=>`$\left[ \begin{array}{l}3x=2\\-x=-2\end{array} \right.$
`<=>`$\left[ \begin{array}{l}x=\dfrac{2}{3}(TM)\\x=2(KTM)\end{array} \right.$
Vậy: GTNN của `A` là: `8` khi `x=2/3`
