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 $A_k,B_k,...$?