Question

x,y>0 x+y>=3

cm

\(x+y+\frac{1}{2x}+\frac{2}{y}>=\frac{9}{2}\)

dau = xay ra khi nao

Answer 1

\(x+y+\frac{1}{2x}+\frac{2}{y}=\frac{x+y}{2}+\frac{x}{2}+\frac{y}{2}+\frac{1}{2x}+\frac{2}{y}=\left(\frac{x}{2}+\frac{1}{2x}\right)+\left(\frac{y}{2}+\frac{2}{y}\right)+\frac{1}{2}\left(x+y\right)\)

Vì x\(\ge0\)  => \(\frac{x}{2}\ge0;\frac{1}{2x}\ge0\). Áp dụng bđt cô si cho 2 số dương ta có:

             \(\frac{x}{2}+\frac{1}{2x}\ge2\sqrt{\frac{x}{2}\cdot\frac{1}{2x}}=2\sqrt{\frac{1}{4}}=2\cdot\frac{1}{2}=1\)

Chứng minh tt ta có:

             \(\frac{y}{2}+\frac{2}{y}\ge2\)

=> \(x+y+\frac{1}{2x}+\frac{2}{y}\ge1+2+\frac{1}{2}\cdot3=\frac{9}{2}\)