若a>0,b>0,a+b=1,则(1/a^2-1)(1/b^2-1)的最小值是

若a>0,b>0,a+b=1,则1/a>1,1/b>1
1=a+b≥2ab^(1/2)
ab^(1/2)≤1/2
ab≤1/4
1/ab≥4 当a=b=1/2取等号
(1/a^2-1)(1/b^2-1)=(1/a+1)(1/a-1)(1/b+1)(1/b-1)
=(1/a+1)(1/b+1)(1/a-1)(1/b-1)
=(1/ab+1/a+1/b+1)(1/ab-1/a-1/b+1)
=(1/ab+(a+b)/ab+1)(1/ab-(a+b)/ab+1)
=2/ab+1≥2*4+1=9
当a=b=1/2
(1/a^2-1)(1/b^2-1)的最小值是为9