# Intermediate Value Theorem

• Jan 25th 2011, 08:33 PM
joatmon
Intermediate Value Theorem
Can somebody help me set this one up, please? I can't quite see how the intermediate value theorem can be directly applied to this.

"A person walks up to the top of a mountain and camps for the night. On the next day, he returns to his car following the same path that he took the previous day. On each day, he starts and finishes his hike at the same time. Use the intermediate value theorem to show that there is a point on the path that the person will cross at exactly the same time of day on both days."

Thanks!
• Jan 26th 2011, 12:05 PM
awkward
Say the length of the path is L and the person starts at time 0 and ends at time T each day.

On the first day, let the distance from the base of the mountain to his location at time t be f(t), so f(0) = 0 and f(T) = L.

On the second day, let the distance from the base of the mountain to his location at time t be g(t), so g(0) = L and g(T) = 0.

f and g are continuous. What can you say about the function f-g?
• Jan 26th 2011, 01:43 PM
joatmon
Thanks for the reply. That's how I ultimately solved it. By subtracting the two functions, you get a single, continuous function that IVT can be applied to.

I appreciate the help!