What is the maximum product of any set of positive real numbers which add to the positive real number N?

this is tricky





my first thought is





e^(N/e)





After thinking


I Know that e^(N/e) is the upper bound





2^(N/2) %26lt; X(N) %26lt;= e^(N/e)





1 X(1) = 1


2 X(2) = 2


3 X(3) == 3


4 X (4) == 4


5 X(5) %26gt;= 6.25 = (2.5)^2


6 X(6) %26gt;= 9 = 3^3


e^(6/e) = 9.0909248530948745926435491751205


9 X(9) %26gt;= 27


e^(9/e) = 27.410193521540484491870166519883


10 X (10) {%26gt;= 39.0625 } = [X(5)]^2


e^(10/e) = 39.598625644627640769306050461851








if n = x e xE I


then X(n) = e^x this is trueWhat is the maximum product of any set of positive real numbers which add to the positive real number N?
Product of sets of how many numbers is necessary ,must be known. If its only 2 numbers i. e. a+b = n, then ab is max when a = b. But for 3 or 4 ...the things will be different.