How to find solution of a equation in n+1 variables so as to find extremal points

Question

Need help in finding maxima of a function .

assume $x_1$ , $x_2$ ,...,$x_{n+1}$ all belong to [0, 1] .

assume r to be any positive integer -> Then what are the solutions of equation.

r- $\frac{ x_l}{1-x_l} $+ (2r+1)× $\frac{ x_1× x_2× ... × x_{n+1} } { 1-x_{1}× x_{2} × x_{n+1} }$ =0.

Edit 1 -> So, the question is prove that the maximum occurs at the diagonal $x_{1}$ = $x_{2}$ = ... =$x_{a+1} $ .

Answer 1

Hope that this is useful for your questions.

For system of equation, you can deduce that $\frac{x_i}{1-x_i}=\frac{x_j}{1-x_j}$ for all $1 \leq i,j\leq n+1$ so $x_i=x_j$ for all $i,j$.

For finding argmax by taking derivative. Since the objective function will be $0$ if any $x_i=0$ or $x_i=1$ (of course we need to assume that $\prod x_i <1$). So we just need to solve the maximization problem under open set $(0,1)^{n+1}$ which has the necessary condition is that the derivative equals to 0.

Answer 2

You would expect to be able to pick $n$ of the variables over some range and calculate the last one. There is not necessarily a nice description of the range for each, as it depends on the other ones.