设a,b为正数,且a+b=1,则1/2a+1/b的最小值是

a+b
=a+b/2+b/2
≥3·(a·(b/2)·(b/2))^(1/3)
=(3/4^(1/3))·(ab^2)^(1/3)
(3/4^(1/3))·(ab^2)^(1/3)≤1,则
1/(ab^2)^(1/3)≥3/4^(1/3);

∴1/2a+1/b
=1/2a+1/2b+1/2b
≥3·(1/(2a·2b·2b) ^(1/3)
=(3/2)/(ab^2)^(1/3)
≥(3/2)·(3/4^(1/3))
=9/2^(5/3)

用柯西不等式的性质得(a+b/2)=(a+b)*(a+b/2)大于或等于(根号a*根号1/a+根号b*根号2/b)的平方故最小值为3/2+根号2