不等式题目：a+b+c小于等于3,求证 1/(a+1) +1/(b+1) +1/(c+1)大于等于3/2

题目

不等式题目：a+b+c小于等于3,求证 1/(a+1) +1/(b+1) +1/(c+1)大于等于3/2

答案

这里有个前提：a,b,c都大于-1
首先用不等式1/(a+1)+1/(b+1)+1/(c+1)≥3/{[(a+1)(b+1)(c+1)]^(1/3)}
而(a+1)(b+1)(c+1)≤[(a+1)+(b+1)+(c+1)]³/27≤6³/27=8
而这是分母,所以3/{[(a+1)(b+1)(c+1)]^(1/3)}≥3/[8^(1/3)]=3/2