Proof of an (Elementary) Inequality Involving Elementary Symmetric Polynomials Let $x_1,\dots, x_n$ be $n$ variables.
For brevity I'll use the following notation: for $I \subset \{1,\dots , n\}$ define
$$
x_I := \prod_{i \in I} x_i.
$$
Let $e_k(x_1,\dots, x_n)$ be an elementary symmetric polynomial. That is
$$
e_k(x_1,\dots, x_n) = \sum_{\substack{I \in \binom{[n]}{k}}} x_I.
$$
Suppose that we have the following conditions on $x_1,\dots, x_n$:
$$
\sum_{i=1}^n x_i = 1 \qquad \text{and} \qquad x_i \geq 0
$$
I have been trying to prove the following inequality without Lagrange multipliers:
$$
e_k(x_1,\dots x_n) \leq \frac{\binom{n}{k}}{n^{k}}
$$
I have had two difficulties:
(1) I have made an (albeit feeble) attempt using AM-GM. Any hints on how one might proceed without Lagrange Multipliers would be very much appreciated.
(2) Additionally, when one does use Lagrange Multipliers I have been having trouble arguing that the maximum does indeed occur when
$$
x_1 = \cdots = x_n = \frac{1}{n}.
$$
Thanks in advance for your help.

Your inequality follow immediately from the Maclaurin's  inequality.
The Maclaurin inequality we can prove by the Rolle's theorem.
