On the Discrete Optimization of x ^(n−x) and its Upper Bounds via the Lambert W Function

Abstract

This paper investigates the discrete optimization of the exponential function g(x) = x^(n−x) subject to a fixed integer n While the continuous relaxation of this problem can be approached via classical differential calculus, determining the absolute maximum over the domain of natural numbers requires a specialized discrete mapping function Utilizing the principal branch of the Lambert W function, we establish a closed-form expression for the exact integer sequence m(n) that yields the maximum value Furthermore, we rigorously prove the concavity of the continuous model analyze the structural limitations of the proposed discrete rounding function for small boundary values (n ≤ 5) and provide both mathematical and graphical proofs that for all valid domain values, the optimal base m(n) is strictly bounded by n/2