Symbolic Representation of Minimum

Question

(2x - c)**2 has a minimum at x = c/2. However, I have a much more complicated algebraic expression that I need to find the minimum of. And I need that minimum in terms of symbols, not an actual number. Specifically:

m(1-n)/(m(1-n) + (1-m)n) - x/(x(1-y) + (1-x)y)

where m = (x + (1-x)(1-c)) and n = (y + (1-y)(1-c)) for some constant c. x, y are both in [0, 1].

What would be a good way to go about finding this? I'd like to find value of x that maximizes this in terms of y and c. Even for y=0 would still be really great. I was trying to use Wolfram-Mathematica but it was not cooperating.

Answer 1

I got confused, your title says minimum and the text says maximum. You can indeed find a maximum :

 Maximize[{expr, 0 <= x <= 1, 0 <= y <= 1}, {x, y}]

..This takes a few minutes..

oddly, the values {0,1/2,Infinity} are correct, however the x,y locations are not. It should be x==0 OR y==0 for c==1 and x==0 AND y==0 for c<1 .. Evidently Maximize is not very good at describing a solution which is not unique.