math extra cerdit?

Question

Among all collections, S, of positive integers whose sum is 28, what is the largest product that the integers in S can form?

Answer 1

2+3+4+5+6+8=28
2x3x4x5x6x8=5,460


if they have to be unique (otherwise 2^14)

Answer 2

in all collection, only integers exist. now we know that for any set of n integers, Arithmetic Mean of those integers >= geometric mean of those integers. let a1, a2....an be n such integers forming a collection in S
then we have (a1+a2+...+an)/n>=(a1.a2.a3....an)^1/n
given a1+a2+a3+...+an=28. let a1.a2.a3...an = x. we need to maximize x
we have x^1/n<=28/n that is x<=(28/n)^n for all n. maximum x means maximum right hand side. now if u have studied calculus, to maximize the RHS, take its derivative with respect to n and equate to 0 ie d((28/n)^n)/dn=0 and find n. to differenciate right hand side, take logarithms and then use product rule.

I have a feeling i went wrong sumwhere but im pretty much sure this is the right way to go about the solution
:)

Answer 3

What is cerdit? Do you do ANY of your homework yourself?

Answer 4

Hi. s^s^s...