I have two equations. In this example I'll make them linear, and later see if the solution generalises to different forms of equation:
$$y_1=-0.1x + 0.65$$ $$y_2=0.15x - 0.825$$
I'm interesting in small ranges of these graphs, say 0.5-2.5 of $y_1$ and the same range +7 of $y_2$. We assume the value of the equation outside of its range is 0.
I want to find a value for x such that the summed integration of $y_1$ from that point up to the end of its valid range, and the integration of $y_2$ from $(x+7)$ down to the start of its valid range, is minimal.
I hope that makes sense. I've sort of fried my brain getting the problem into these terms, and now can't see the solution for the trees.