Q: Piecewise linear function given three points and two crossover boundaries Suppose you have three points; (3500, 700), (52500, 5075), and (527500, 36800).  As well as two x boundaries 25000 and 200000.  The question is then to construct three lines (each of which goes through one of the points) in such a way that the lines achieve the same value at each boundary point.  The catch is that the second line must have a lesser slope than the first line and the third line must have a lesser slope than the second line.  
I do not believe a solution exists but I would like to know how to set up the problem in order to check for solutions in r.  Ideally, I would like to then relax the constraint that the functions be equal at the boundaries to something where they are within 10% of one another or something like that.
Edit 1:
Essentially what I want is a line that goes through (3500, 700) and has a value (25000, y_1), a line that goes through (52500, 5075) and has values (25000, y_1) and (200000, y_2), and a third line that goes through (527500, 36800) and has a value(200000, y_2).
Edit 2:
Here is an updated graphic.  The dashed lines represent an incorrect solution because the slope of the last segment is larger than the slope of the middle segment (which is 0).  What I am essentially looking for is the y = mx + b format of each of the three segments if they were infinitely extending lines.  

