Can someone help prove my hunch?

[Archosaur]

I have this feeling that
if the sum of the finite series a+b+c+d...+z= some constant C
then the product a*b*c*d...*z has a maximum when a=b=c=d...=z

For example, if C=20, and you just use a,b,c,and d then...
6*6*6*2=432
3*8*8*1=192
5*5*5*5=625 (Maximum)

Can someone prove / disprove this?

 

[Filip Larsen]

No doubt this is a fairly standard optimization problem, but taking this as an opportunity to think about it before rushing to a reference, you can perhaps look at just two elements and start with them being equal, that is

a + a = C,
a * a = P

and then add an amount da > 0 to the first element and subtract it from the other

(a + da) + (a - da) = a + a = C
(a + da) * (a - da) = a² - da² < P

So for two elements, having the same value is obviously a (local) maximum for the product and you only need to prove it is global and extend to multiple elements.

 

[Archosaur]

(a + da) * (a - da) = a² - da² < P

Thanks, this is totally clear to me now. I started doing something like this, but... well... I don't know if it's 4:00 am where you are, but it is, here. Haha, thanks.

 

[Filip Larsen]

Note, that if the elements are allowed to be negative (which I presume they are not in your case) it is easy to see that the product can be made arbitrary large for the same sum. For instance, if we take a + b + b = C with an arbitrary a > C this implies that b = (C-a)/2 < 0 and the product P = a*b*b must go to infinity as a goes to infinity. Disallowing negative elements ties into proving that the solution for the maximum product is a global maximum.