Question

已知a.b.c为正实数.且a+b+c=1,求证:(1/a-1)(1/b-1)(1/c-1)>=8

Answer 1

a.b.c为正实数.且a+b+c=1
(1/a-1)(1/b-1)(1/c-1)
=[(1-a)/a]*[(1-b)/b]*[(1-c)/c]
=[(b+c)/a]*[(a+c)/b]*[(a+b)/c]
≥[2√bc/a][2√ac/b][2√ab/c]
=8abc/abc
=8
得证