若a>0,b>0.求证a+b+1/根号ab大于等于2根号2麻烦写下过程

Question

若a>0,b>0.求证a+b+1/根号ab大于等于2根号2
麻烦写下过程

Answer 1

a>0,b>0,a+b>=2(ab)^(1/2),2(ab)^(1/2)代表2乘以根号ab.
a+b+1/(ab)^(1/2)>=2(ab)^(1/2)+1/(ab)^(1/2),设(ab)^(1/2)=x,所以a+b+1/根号ab>2x+1/x.
x>0,而2x+1/x>=2(2)(1/2).a+b+1/(ab)^(1/2)>=2(2)(1/2).