19 Aug '08 15:45>
Which value of x between 0 and 1 maximizes x^m*(1-x)^n+x^n*(1-x)^m, where m and n are positive integers?
Originally posted by David113My first thought was to say that since this function is symmetrical with respect to x and (1-x), the answer would either be an extreme point or the mid-point. The extreme values x=0 and x=1 make the function f(x) equal to 0. However, after graphing this function using various values of m and n, I found that certain combinations create two maximums equally spaced from the centre.
Which value of x between 0 and 1 maximizes x^m*(1-x)^n+x^n*(1-x)^m, where m and n are positive integers?
Originally posted by PBE6For x=0 and x=1 the function (let's call it F) is at a minimum, so for the rest of the train of thoughts I will assume x is strictly between 0 and 1.
My first thought was to say that since this function is symmetrical with respect to x and (1-x), the answer would either be an extreme point or the mid-point. The extreme values x=0 and x=1 make the function f(x) equal to 0. However, after graphing this function using various values of m and n, I found that certain combinations create two maximums equally spa ...[text shortened]...
The derivative of the function is a little ugly, so I'm going to look for a simplification.