What is the lowest value of a product of two positive numbers a and b, given that a+b=a*b?

(In reply to Analytical solution (spoiler) by Steve Herman)

Taking off from b=a/(a-1):

y = ab = a^2 / (a-1)

y' = (a^2 - (a-1)*2*a) / (a-1)^2

which is zero when

a^2 = 2*a^2 - 2*a

a^2 - 2*a = 0

a=0 or a=2

a=0 is not positive, and also for y itself 0<a<1 makes b negative.

a=2 can be shown to be a minimum for y rather than a maximum, so b = 2 and ab=4.

Posted by Charlie on 2013-04-12 14:04:01