need help with a calculus problem involving the intermediate value theorem?

Question

A backpacker leaves the trailhead on Monday at 9am. and walks a trail arriving at hidden lake at 3pm. his journey may have included rest stops, lunch, a nap, and returning to a stream crsossing where fisheing rods were left after getting a drink. on tuesday, the backpacker leaves hidden lake at 9am. and returns along the same trail. we aren't given the time of arrival at the trailhead. Prove: The backpacker was at some location at the exact same time on both monday and tuesday.

Answer 1

If x = m(t) is the backpacker's position at time t on Monday and u(t) is his position at time t (in hours ater 9 am) on Tuesday, where x is an index of position along the continuous path, we know that m(0) = 0, m(6) = 1, and u(0) = 1. Also, there exists a time b > 0 such that u(b) = 0. We seek to prove that m(t) and u(t) have an intersection, which is to say that there exists a time c such that m(c) = u(c), and let us say that m(c) = u(c) = w. By the intermediate value theorem, both m(t) and u(t) assume all values for x in the interval [0, 1]. If the intersection does not exist, it raises an impossibility. u(0) = 1 and m(0) = 0, so u(0) > m(0). u(t) must assume all values between 1 and any future value, so if u(t) never intersects m(t), it must remain true that u(t) > m(t) for any t. But we know that u(b) = 0 and m(6) = 1. Suppose b < 6. Then m(b) >= 0, so u(b) > m(b) fails. Suppose b = 6. Then m(b) = 1, so u(b) > m(b) fails. Suppose b > 6. Then u(6) <= 1, so u(6) > m(6) fails. Thus, the intersection must exist.

Answer 2

I don't get it, you lost me at the second line. I would hate to have your homework!!!!!!!! Tell your teacher you went over it numerous times but still couldn't find the answer.