I am trying to find the set of $x_{i}$ that maximize $\prod_{i=1}^{N}{\left(1+x_{i}\right)}$ given that $\prod_{i=1}^{N}{x_{i}} = c$ and $0\leq x_{i} \leq 1$ for all $i$. As far as I can tell unless the solution set is $x_{i}$ is equal to $\sqrt[n]{c}$ for all $i$, then there is not going to be a unique solution, so an extra constraint along the lines of $x_{i} \leq x_{i+1}$ is also fine.
For $N$ equal to 2, the solution appears to be $x_{i}$ is equal to $\sqrt[n]{c}$ for all $i$. I get this by expanding the product, taking the derivative, then solving for when the derivative is equal to zero, and then confirming it is a maximum with the second derivative. But apparently my math is rustier than I thought, and I have in fact confirmed that $\sqrt[n]{c}$ is a minimum since the second derivative is positive.
I might be able to use the same approach for $N$ equal to 3, but there are more terms and you need to take partial derivatives. The problem is it doesn't get me closer to a general solution.
Using the logarithm hint makes everything look nicer. The problem then become to maximize $\sum_{i=1}^{N}{\log{\left(1+x_{i}\right)}}$. I think I want to find where all the partial derivatives are equal to zero. The partial derivative is $\frac{1}{1+x_{i}}$ which is never zero. I think this means my maximum lies on the boundary, but I don't understand what a multivariate boundary is.