The position function for the trip up is .

The time taken for run up the mountain is minutes.

The position function of trip up at the foot of the mountain is .

The position function of trip up at the top of the mountain is .

The position function for the trip down is .

The time taken for run down the mountain is minutes.

The position function of trip down at the top of the mountain is .

The position function of trip down at the foot of the mountain is .

The function is .

Find .

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The elevation at the foot of the mountain is lower than at the top of mountain.

.

Find .

The elevation at the half way point up of the mountain is higher than the foot of mountain.

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The position functions and are continuous functions.

If two functions and are continuous, then also continuous.

Intermediate value theorem:

If is continuous on the closed interval , , and is any number between and , then there is at least one number in such that .

and .

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By intermediate value theorem, there must be some in , such that .

.

.

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is the time between and .

At time he passed the same place at same time on Saturday. 

At time he passed the same place at same time on Saturday.