已知a+b=1且a>0,b>0,求证(a+1/a)(a+1/b)>=9
2个回答

转化为两个题目:

已知a>0,b>0且a+b=1,求证(1+1/a)(1+1/b)>=9

已知a>0,b>0且a+b=1,求证(a+1/a)(b+1/b)>=25/4

下面证明:(a+1/a)(b+1/b)>=25/4

(a+1/a)(b+1/b)

=ab+b/a+a/b+1/(ab)

=(a^2b^2+b^2+a^2+1)/(ab)

=[a^2b^2+(a+b)^2-2ab+1]/(ab)

=[a^2b^2+1-2ab+1]/(ab)

=a^2b^2/ab-2ab/ab+2/ab

=ab+2/ab-2

1=a+b>=2√(ab)

所以√(ab)