已知a,b,c属于正实数,a+b+c=1,求(1/a-1)(1/b-1)(1/c-1)大于等于8

题目

已知a,b,c属于正实数,a+b+c=1,求(1/a-1)(1/b-1)(1/c-1)大于等于8

答案

证明：∵ a+b+c=1∴ (1/a-1)(1/b-1)(1/c-1) =[(a+b+c)/a-1]*[(a+b+c)/b-1]*[(a+b+c)/c-1] =(b/a+c/a)*(a/b+c/b)*(a/c+b/c) ≥2√(bc/a²)* 2√(ac/b²) *2√(ab/c²) =8∴ (1/a-1)(1/b-1)(1/c-1)大...