1. The problem statement, all variables and given/known data Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f(t)=g(t)-h(t), where g and h are the position functions of the two runners.] 2. Relevant equations If this is ever helpful: Mean Value Theorem: f ' (c)= [f(b)-f(a)]/[b-a] 3. The attempt at a solution We know that if a function f is continuous on an interval [a,b] and differentiable on (a,b), and f(a) = f(b) = 0, then there is some point c in (a,b) such that f'(c) = 0. The velocity equation is f'(t)=g'(t)-h'(t) g^' (t)-h(t)=0 g'(t)=h'(t) I am not at all sure I am doing it correctly - my solution is too simple.