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 .

.

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.

.

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 .

.

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

.

.

 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.