I would put this question in the University Math Help\Advanced Applied Math or University Math Help\Other Advanced Topics section. Just a heads-up.

I have a couple of questions: presumably you're trying to find CMC, or the intersection of these two fitted lines. Do you have any bounds on the location? For example, do you always know that this many data points corresponds to the first line, and the next however many data points corresponds to the second line? Or perhaps do you always know that the CMC occurs after a particular value of m or before a particular value of m?

Or here's another question: do you always know that the function is concave down?

Here's one algorithm:

1. Take clusters of 5 contiguous data points (or so). Fit parabolas to them.

2. Take the second derivatives of each of the fitted polynomials (this equals twice the coefficient of the term).

3. Find the most negative of these second derivative values. That is your CMC.

Here's another algorithm:

1. Rotate all of your data points through an angle such that the y-component of the first data point is equal to the y component of the last data point.

2. Run a peak-finding routine to find the maximum value of the rotated data points. Keep track of the index of the data point corresponding to this maximum.

3. Rotate all the data points through an angle to get back to your starting point. The x and y component of the indexed data point is fairly close to the CMC. You'd have to analyze the possible error here, because of the resolution of your data points.