Cho x, y > 0 thỏa mãn x + y [tex]\geq[/tex] 3. Chứng minh: x + y + [tex]\frac{1}{2x}[/tex] + [tex]\frac{2}{y}[/tex] [tex]\geq \frac{9}{2}[/tex]
$x+y+\frac{1}{2x}+\frac{2}{y}$
$=(\frac{1}{2x}+\frac{1}{2}x)+(\frac{2}{y}+\frac{1}{2}y)+(\frac{1}{2}x+\frac{1}{2}y)$
$\geq 2\sqrt{\frac{1}{2x}.\frac{1}{2}x}+2\sqrt{\frac{2}{y}.\frac{1}{2}y}+\frac{1}{2}(x+y)$ ( BĐT Cauchy)
$\geq 1+2+\frac{1}{2}.3$
$=\frac{9}{2} (đpcm)$