# Word Problem : Intermediate Value Theorem

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At 8am on Saturday, a man begins running up the side of a mountain to his weekend campsite. On Sunday at 8am, he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on his way down, he realizes that he passed the same exact place at exactly the same time on Saturday. Prove that he is correct.(hint:Let u(t) and d(t) be the position functions for the runs up and down, and apply the intermediate value theorem to f(t)=u(t)-d(t).

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##### Solution Summary

The Intermediate Value Theorem is applied to solving a relative time and distance problem.

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Ok, we consider f(t)=u(t)-d(t) where u(t) shows the position function for run up and d(t) shows the position function for run down. We should define a direction and stick to it till the end. Let's say the start point is the origion ...

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