for what arguments the function reaches maximum? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:23:00Z http://mathoverflow.net/feeds/question/123567 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123567/for-what-arguments-the-function-reaches-maximum for what arguments the function reaches maximum? Chris 2013-03-04T20:10:27Z 2013-05-25T07:22:00Z <p>Hi,</p> <p>What is the maximum of the following function?:</p> <p>$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ x_i}\right )}$</p> <p>given:</p> <p>$i = 1,..,n$</p> <p>$1 &lt; w_i$</p> <p>$0 &lt; x_i &lt; 1$</p> <p>The generalization of the <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means" rel="nofollow">inequality of arithmetic and geometric means</a> 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.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/123567/for-what-arguments-the-function-reaches-maximum/123667#123667 Answer by Hans for for what arguments the function reaches maximum? Hans 2013-03-05T21:11:12Z 2013-03-06T13:05:44Z <p>For $n=1$ you get $f=0$ for all values.</p> <p>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 $f= \infty$.</p> <p>EDIT:</p> <p>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 $f= \infty$.</p>