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edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty.

Edit 3: Can somebody answer?

Edit 4:I have been thinking this about for a month, but generating nothing, can somebody answer this?

edit 3: The 100 point bounty is already end, so following the rules, i have to create a 200 points bounty. If you love this question, please upvote my question and show some support by answering.

This is the easier version of this problem:

What is a working example and a non-working for finding the maximum of a special function $f(a,b,c,d,e)$ with 4 equality / inequality constraints?

problem no. 50: Part a) What is the general method to find all the maximum/minimum of a algebraic equation and a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 constraints?

If there is no finite numbers of method to solve all maximization problems." rigourously and formally?

The function itself can either be a non-linear algebraic function. Each nonlinear algebraic(no linear here) constraints and must use more than one variable and are some inequality (appear more than 1 times) and some equality(appear more than 1 time). Assuming there are no closed form at the end, could you provide the exact infinite sum method?

Appreciate your motivating example in advance!

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I think the reason no one seems to be answering your question is because it is very unclear what you are asking for. Are you asking for a worked out example of how to find the maximum of a function of 4 variables? If so, what do you mean by each constraint must include some equality and some inequality? And is there a particular method of solving for the maximum you are looking for? E.g. Lagrange multipliers, numerical simulations, pure analysis? –  nullUser Oct 1 '12 at 21:07
    
E.g. I could just take $f(x,y,z,t,s) = 1-x^2-y^2-z^2-t^2-s^2$. Take the constraints $x^2+y^2 = 0, y^2 + z^2 =0, z^2 + t^2 \leq 1, t^2 + s^2 \leq 1$. Then the maximum is clearly at $x=y=z=t=s=0$. But I don't think this is what you are looking for. I recommend cleaning up your question and being more specific about what you want. –  nullUser Oct 1 '12 at 21:11
    
@nullUser - The general rule of thumb that could address all types of algebraic equation –  Victor Oct 1 '12 at 21:14
    
There is no one method to solve all maximization problems. You have to give some context to the problem. –  nullUser Oct 1 '12 at 21:15
    
if you are referring to the example I gave in my comment above, yes I can. Note that $-x^2-y^2-z^2-t^2-s^2$ is a nonpositive quantity. It follows that $f(x,y,z,t,s) \leq 1$. Plug in all zeros and you get $f(x,y,z,t,s) = 1$. Note that all zeros satisfy the constraints as well. It follows that this is a global maximum of $f$. –  nullUser Oct 1 '12 at 21:23

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