@Markiyan Hirnyk 

Dear Markiyan,

Since you took the time to think about this problem and help me, I think I should ask my problem in full instead of just part of it. I believe, then, you will perhaps agree that the issue at hand is not a prejudice against a procedure. By all means, I am interested in solving the problem with ANY method. In my humble opinion, the problem is, to say the least, not trivial. Let me rephrase: I have not been able to do it yet.

Thank you for your time again.

The problem:

*** f is a function of x and y ( a and b are constants for the fucntion f)

***g is a fucntion of a and b (x and y are constants for the function g)

***Maximize g with respect to a and b,

where x and y are such that they minimze the fucntion f.

*** constarints: 0<a<x<100, 0<b<y<100,

f=(-31250000 + 312500 (x + y) +
 625 (4 a x - 4 x^2 - 5 x y + 4 (b - y) y) +
 25 x (-a b + b (x - y) + y^2))/((-500 + 5 a) (500 - 5 b))

g=(-1/(1200 (100 - a) (-100 + b)))(-2 a^3 (-100 + b) + 200 b^3 - b^4 +
   6 a (-100 + b) (200 - x) x + 120000 x^2 - 800 x^3 + x^4 +
   120000 y^2 - 1200 x y^2 - 800 y^3 + 8 x y^3 +
   6 b (x^3 - 20000 y - x y^2 + 100 (-2 x^2 + 2 x y + y^2)))

I will save that screen shot :)

I am trying by hand...not automatic

I will save that screen shot :)

I am trying by hand...not automatic

@Markiyan Hirnyk 3453 Thank you for your help and insight...

I was interested in getting symbolic solutions as I will have to use them in a later part of the problem. Having said that, I would even be interested in the solution of the same problem where M=100 for example. Even in this case, I am not sure if I will be able to get a symbolic solution.

Your gut feeling on this question: Do you think one can get an answer from Mathematica on a main frame for example, a much faster computer? Or is it the case that not having an answer in 1 hour on your computer shoulc mean there probably is not a symbolic solution found? Thank you for your time again.

@Markiyan Hirnyk 3453 Thank you for your help and insight...

I was interested in getting symbolic solutions as I will have to use them in a later part of the problem. Having said that, I would even be interested in the solution of the same problem where M=100 for example. Even in this case, I am not sure if I will be able to get a symbolic solution.

Your gut feeling on this question: Do you think one can get an answer from Mathematica on a main frame for example, a much faster computer? Or is it the case that not having an answer in 1 hour on your computer shoulc mean there probably is not a symbolic solution found? Thank you for your time again.