Question

use the intermediate value theorem to explain why the hands of the clock coincide at least once between 1pm and 1:15pm

Answer 1

Let f(x) be the angle (x) between the minute and the hour hand:


Then f(x) > 0 after 1:00 pm and f(x) < 0 at 1:15 pm. The Intermediate Value Theorem says says that somewhere in between f(x) = 0. This is where the two hands coincide.

Does this make sense?