When logic and calculus collide

[GRaduns340]

It makes sense.

I'm not sure your assumption of the starting velocities being the same is completely true though. Technically, there is no derivative of the position graph at zero, because the derivative doesn't exist at the endpoints of a graph. Logically, yes, their velocities are both 0, but in order to use a derivative to prove it you would have to assume that the runners were both at rest before the start of the race, but with a lack of that knowledge there is no way to find f'(0).

You'd actually have to prove that it was an intermediate time at which their velocities were the same, like Tim suggested. I don't know if you've done it yet, but the mean value theorem would be useful there. (I think I pulled out the right name).