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Suppose I have two rational polynomials $f_1 (x)$ and $f_2 (x)$, both of the form $$ f_k (x) = \frac{x (A_k x + B_k)}{(C_k x + D_k)} $$ with constants $A_k, B_k, C_k$ and $D_k$ and where $k=1,2$. I am interested in finding at what $x$ value(s) the function $$ g (x) = \min[x, f_1(x)] + \min[f_2(x), 1 - x] $$ has a minimum in the range $x \in [0, 1]$; the denominators of both $f_k(x)$ are positive in this range.

Is there a way to do this without explicitly solving each of the four possibilities for their minima and comparing? In other words, based on the relative sizes of the constant parameters $A_k,B_k,...$?

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This question looks likely to be closed. If so, there are other mathematical Q&A sites you can ask this at, listed in the FAQ. Please don't post this question at other sites immediately, see what happens here first for a day or so. –  David Roberts Oct 11 '11 at 11:14
    
On a different note, we write x\in[0,1], not x=[0,1]. Is there any particular reason you are interested in this question? In what context did it crop up? –  David Roberts Oct 11 '11 at 11:16
    
It seems unlikely to me that it would be easier to evaluate this question "qualitatively" as you are asking than to first solve it in the standard case-by-case method and then see what the answer implies can be said qualitatively. Random complicated questions usually don't admit simple uncomplicated answers. –  Jack Huizenga Oct 11 '11 at 14:43
    
I changed the "=" to a "\in" -- I had previously sent this as a text file to friends, but I guess I missed changing that one... The context of this problem is that g(x) gives the numerical change in a minimization problem; the mins and maxs come from the fact that you cannot gain or lose more than the original number. I can provide more details if you wish... I did not expect much, but I thought I'd post it and see if there were methods I was unaware of. –  Daniel Cartin Oct 15 '11 at 21:26
    
You might consider computing the derivatives of the f_i, so that you have an idea when to check what endpoints. Personally, given specific values for the coefficients, I would plot points. Gerhard "Ask Me About Point Plotting" Paseman, 2011.10.15 –  Gerhard Paseman Oct 16 '11 at 5:29
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