Question

已知 abc属于正实数,且a+b+c=1 求证(a+1/a)(b+1/b)(c+1/c)≥1000/27

Answer 1

1)a+b+c=1≥3(abc)^1/3
abc≤1/27 1/abc≥27
(1/a+1)(1/b+1)(1/c+1)
=1/a+1/b+1/c+1/ab+1/bc+1/ac+1+1/abc≥3(1/abc)^1/3+3
(1/abc)^2/3+1/abc+1=64 >=1000/27