**
Problem**

At a particular horse race, two horses start at the same time, and finish in a tie. Show that at some time during the race, the horses were running at the same speed.

**
Solution**

By carefully choosing our functions, we can solve this using Rolle's Theorem from section 2.10 in the textbook.

Let f(t) and g(t) be the distances traveled at time t by the two horses. The functions f '(t) and g'(t) are therefore the speeds of the horses at time t. Define h(t) = f(t) – g(t). Let b be the finishing time of the race. Translated into math notation, the problem states that

implying

Thus by Rolle's Theorem, there exists a time c, 0 < c < b such that h'(c) = 0. Since h(t) = f(t) – g(t), this means f '(c) – g'(c) = 0, or f '(c) = g'(c); that is, at time c the two horses are running at the same speed.