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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.]

Let $g(t)$ and $h(t)$ be the position functions of the two runners and let $f(t)=g(t)-h(t) .$ By hypothesis,

$f(0)=g(0)-h(0)=0$ and $f(b)=g(b)-h(b)=0,$ where $b$ is the finishing time. Then by the Mean Value Theorem,

there is a time $c,$ with $0<c<b,$ such that $f^{\prime}(c)=\frac{f(b)-f(0)}{b-0} .$ But $f(b)=f(0)=0,$ so $f^{\prime}(c)=0 .$ since

$f^{\prime}(c)=g^{\prime}(c)-h^{\prime}(c)=0,$ we have $g^{\prime}(c)=h^{\prime}(c) .$ So at time $c,$ both runners have the same speed $g^{\prime}(c)=h^{\prime}(c)$

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