Find local extrema $f(x_1,x_2, \ldots , x_n) = \sqrt{(x_1+x_2+\ldots x_n-a)(a-x_1)(a-x_2)\cdots (a-x_n)}$

Question

Find local extrema of the following function of $n$ variables, $n \ge 2$ within its domain: $f(x_1,x_2, \ldots , x_n) = \sqrt{(x_1+x_2+\ldots x_n-a)(a-x_1)(a-x_2)\cdots (a-x_n)}$

I have rewritten the function into products and sums: $f(x_1,x_2, \ldots , x_n) = \sqrt{\left(\sum_{k=1}^{n}x_k-a\right)\prod_{k=1}^{n}(a-x_k)}$

The partial derivative of f with respect to $x_i$ is: $f'_{i}(x_1,x_2, \ldots , x_n) =\frac{1}{2f(x_1,x_2, \ldots , x_n)}\left(1\cdot \prod_{k=1}^{n}(a-x_k) - \left(\sum_{k=1}^{n}x_k-a\right)\prod_{k=1, k\ne i}^{n}(a-x_k)\right)$

First, I need to find all the stationary points, but this is the point where I got stuck. I know I have the following set of n equations for n variables: $f'_{i}(x_1,x_2, \ldots , x_n) = 0$ where $i \in \{1,\ldots,n\}$, but I don't know how to solve it.

Answer 1

If all $a-x_i\geq0$ then we can use AM-GM: $$f(x_1,...,x_n)\leq\sqrt{\left(\frac{x_1+...+x_n-a+a-x_1-...+a-x_n}{n+1}\right)^{n+1}}=\sqrt{\left(\frac{(n-1)a}{n+1}\right)^{n+1}}.$$ The equality occurs for $x_1=x_2=...=x_n$ and $nx_1-a=a-x_1,$

which says that we got a maximal value.

The minimal value is $0$ of course.