已知a>0,b>0,且a+b=1,则(a+1/a)(b+1/b)的最小值为

Question

已知a>0,b>0,且a+b=1,则(a+1/a)(b+1/b)的最小值为

Answer 1

(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+b=1
=[a^2b^2+1-2ab+1]/(ab)
=a^2b^2/ab-2ab/ab+2/ab
=ab+2/ab-2
a+b=1>=2√(ab)
√(ab)<=1/2
∴0<ab<=1/4

∴ab+2/ab-2>=(1/4)+2/(1/4)-2=25/4
(a+1/a)(b+1/b)的最小值为25/4

Answer 2

1.5 可能对。。我自己算的..
注：我数学一般

Answer 3

1.5