Maximizing A Symmetrical Function

The June 1990 Issue of Mathematics Magazine presented several methods
for determining the maximum value of the function

   f(x1,x2,..,xn)   =   x1 x2 ... xn (x1^2 + x2^2 + ... + xn^2)

where x1 + x2 + ... + xn = 1 and each x is non-negative.  Most 
solutions were based on algebraic relations and induction arguments 
beginning with small values of n, but it's interesting to approach
this as an ordinary problem in the calculus of several variables.
Using the slightly more general constraint x1 + x2 +... + xn = s
we can eliminate xn and express f as a function of x1 through xj
where j=n-1 as follows

 f = x1 x2 ... xj [S-x1-x2-...,xj][x1^2 + x2^2 + ... (S-x1-x2-..-xj)^2]

The function is zero on the boundary of the region with all xi greater
than or equal to zero, and finitely positive on the interior, and a
maximum occurs precisely when the partials of f with respect to each
xi vanish, i.e.,


    df           /           \  /                         \
   --- = (xn-xk)(   PROD   xi )( (xn-xk)^2  +   SUM  xi^2  ) = 0
   dxk           \ i=/=n,k   /  \             i=/=n,k     /


for k=1,2,..,n-1.  Since the two right-hand factors are necessarily
positive in the interior, this implies xk = xn for k=1,2,..,n-1.