Let $n$ be a positive integer bigger than one. Find the maximum value of:

$x\_1^3 + x\_2^3 + \ldots + x\_n^3$

given that $x\_1 + x\_2 + \ldots + x\_n = 0$ and $x\_1^2 + x\_2^2 + \ldots + x\_n^2 = 1$.

This is what I have tried:

Analyzing both conditions separately it is possible to conclude that such value is greater than $0$ and smaller than $1$.
Using Newton's identities we have:

$P\_3 = e\_1\ P\_2 - e\_2\ P\_1 + 3 e\_3 = 3 e\_3$ (since $e\_1 = P\_1 = 0$) and $1 = P\_2 = -e\_2/2 \geq e\_2 = -1/2$.

If we take the partial derivative of $S$ with respect to $x\_i$, we get: $3 (e\_2 - x\_i(e\_1 - x\_i)) = 3 ((x\_i)^2 - 1/2)$.

The real problem is that I don't know how to use all this information to solve the problem.