设a>0 b>0,a+b=1求证1a+1+1ab>=8

题目

设a>0 b>0,a+b=1求证1a+1+1ab>=8

答案

1a+1+1ab
=(a+b+1)/ab
=2/ab
又
1=a+b≥2根号(ab)
根号(ab)≤1/2
ab≤1/2
故
1a+1+1ab
=2/ab
≥2 / (1/4)
=8
证毕