AM-HM

來源: 2015-06-05 18:15:12 [舊帖] [給我悄悄話] 本文已被閱讀:

由算術-調和平均不等式。

5/(x+y+1) = 5[1/(x+y+(1/3)+(1/3)+(1/3)] =(1/5)(1/x + 1/y) + 9/5

代入右邊,移項化簡。不等式的證明歸結於證明

1/x + 1/y + 1/z >= 9.

LHS >= 3/cuberoot(xyz) >= 3/((x+y+z)/3) = 9.