已知ab∈R+,a+b=1,求证(a+1/2)^1/2+(b+1/2)^1/2≤2

题目

已知ab∈R+,a+b=1,求证(a+1/2)^1/2+(b+1/2)^1/2≤2
知道的告诉下.谢谢

答案

[(a+1/2)^1/2+(b+1/2)^1/2]^2
=a+1/2+b+1/2+2√[(a+1/2)(b+1/2)]
=(a+b)+1+2√[ab+1/2(a+b)+1/4]
=1+1+2√(ab+1/2+1/4)
=2+2√(ab+3/4)
a>0,b>0
1=a+b>=2√ab
√ab