Say we have $\sum_{i=1}^n x_n = C$, i.e., $x_1+x_2+...x_n=C$ where C is a constant and $x_1,x_2,...x_n$ are nonnegative. Prove the product $(x_1)(x_2)(x_3)...(x_n)$ has a maximum if and only if $x_1=x_2=x_3=...=x_n= \frac{c}{n}$.
I’ve tried plugging in stuff like $x_1=C-x_2-x_3-x_4...-x_n$ into the product for each $x$ value to take the gradient to see if $\frac{c}{n}$ is a critical point but it ends up a really messy derivative that I don’t even know where to start on.
All help is appreciated, thank you.
Hint: Use that $$\frac{x_1+x_2+x_3+...+x_n}{n}\geq \sqrt[n]{x_1\cdot x_2\cdot x_3\cdots x_n}$$