Question

若a.b为正数,a+b=1则(a+1/a)(b+1/b)的最小值是

Answer 1

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

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

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

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

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

=ab+1/(ab)+(1-2ab)/(ab)

=(ab+2/(ab))-2

注意到a+b=1,a