Question

已知a>0,b>0.求证2ab／（a+b）≤√ab≤（a＋b）／2≤√[（a²＋b²）／2]

Answer 1

√[（a²＋b²）／2]

>=(1/2) √[2(a²＋b²)]=(1/2)√[(a^2+b^2)+(a^2+b^2)]>=(1/2)√(a^2+b^2+2ab)=(a+b)/2

>=2√[ab]/2=√[ab]

=2ab/(2√[ab])>=2ab/(a+b)