When logic and calculus collide

[Lil' Lavery]

Originally Posted by Richard

This does not follow from the problem statement. In practice a runner's velocity at the finish line is rarely zero, and in some kinds of racing (relays, sailing, etc.) the starting velocity is not zero, either. But on a purely mathematical level, the statement "the derivative of a constant is zero" is neither precise nor pertinent here.

f'(t) was the difference of the two derivatives, not the derivative of either of the runners positons, and so could (and would have to be) zero. g'(t) and h'(t) were the velocities of the runners, and niether could have been zero (unless it was a race with absolutely no distance or where niether runner moved). The problem did state that both runners started at the same time, but you're right in that it may not mean they both had a velocity of 0 at the beginning.
I get the Calculus behind it (including the mean value theorem, which was the lesson where we were given this problem), I was trying to justify it logically, and I guess that it must be logically true. I've thought about it a little more and at one point they must both have been travelling the same speed.