# Two runners start a race at the same time and finish in a tie. Prove t

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 $f\left(t\right)=g\left(t\right)-h\left(t\right)$ , where and are the position functions of the two runners.
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Steacensen69
Step 1
$f\left(t\right)=g\left(t\right)-h\left(t\right)$ - the difference positions of two runners.
$g\left(t\right)$, $h\left(t\right)$ are continuons and differentiable. So, $f\left(t\right)$ is too continuons and differentiable.
We are attempting to prove that at some point during the race the difference in velocities between the runners was 0.
Find the first derivative of the equation $f\left(t\right)=g\left(t\right)-h\left(t\right)$.
${f}^{\prime }\left(t\right)={g}^{\prime }\left(t\right)-{h}^{\prime }\left(t\right)$
Consider the intreval $\left({t}_{0},{t}_{1}\right)$, where is the start time ${t}_{0}$ and the finish time ${t}_{1}$.
Is given that runners start a race at same time and finis in a tie.
So, $g\left({t}_{0}\right)=h\left({t}_{0}\right)$ and $g\left({t}_{1}\right)=h\left({t}_{1}\right)$.
We have: $f\left({t}_{0}\right)=0$ and $f\left({t}_{1}\right)=0$
We can apply the Rolle's theorem, because we have: $f\left({t}_{0}\right)=f\left({t}_{1}\right)$ and $f\left(t\right)$ is continuons and differentiable.
So, there exists some cc that such ${f}^{\prime }\left(c\right)=0$
${f}^{\prime }\left(c\right)={g}^{\prime }\left(c\right)-{h}^{\prime }\left(c\right)=0$
$⇒{g}^{\prime }\left(c\right)={h}^{\prime }\left(c\right)$
For $c=t$, $t\in \left({t}_{0},{t}_{1}\right)$
$⇒{g}^{\prime }\left(t\right)={h}^{\prime }\left(t\right)$
Apply the Rolle's theorem.