for what arguments the function reaches maximum?

Question

Hi,

What is the maximum of the following function?:

$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ x_i}\right )} $

given:

$i = 1,..,n$

$1 < w_i$

$0 < x_i < 1$

The generalization of the inequality of arithmetic and geometric means may indicate that the maximum is reached, when all $x_i$ are equal (the minuend is a weighted harmonic mean) but I haven't managed to proove it.

EDIT: The above question was badly constructed. The answer I am looking for is: For what values of $w_i$ and $x_i$, in case of any $i=1,..,n$ the following function reaches maximum.

EDIT2: I am looking for the optimum distribution of $w_i$ and $x_i$. The specific values of $w_i$ and $x_i$ are not needed. The problem needs to be solved, knowing that the minuend is a set number: $S=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} }$ . So actually the question can be redefined: for given S, what is the optimum distribution of $w_i$ and $x_i$ so $f(x_i,w_i)$ reaches maximum.

Answer 1

For $n=1$ you get $f=0$ for all values.

For $n=2$ you get for $w_1 = w_2 \approx 1$ and $x_1 = x_2 = 1/2$ division by zero in the second term, and so $f= \infty$.

EDIT:

For every even $n$ you get division by zero for $w_1 = \ldots = w_n \approx 1$ and $x_1 = \ldots = x_n = 1/2$, and so $f= \infty$.