1. ## Interesting Word Problem

My professor gave this word problem to us and I can't seem to navigate through all of the directions and hints to understand how to do it. I do know for sure he expects us to use the Intermediate Value Theorem.

Prove that at any instant of time, there are two antipodal points (two
points whose midpoint is the center of the earth) on the equator with the
same temperature, assuming that temperature changes continuously with
the points on earth.
Hints
Picture the equator as a unit circle and set up a coordinate system. Consider
a point P on the circle. We can represent the point P by the angle theta
between the line connecting the origin to the point P and the positive x-axis.
This way, the points on the equator may be labeled by angles theta in [0, 2pi].

2. Consider $\displaystyle T$ is a continuos function:

$\displaystyle f(0) = T(0) - T(\pi) \neq 0$, thus $\displaystyle f(0) < 0$ or $\displaystyle f(0) > 0$. Now $\displaystyle f(\pi) = T(\pi) - T(0) = -f(0)$ wich will be either positive or negative. In any case, $\displaystyle 0 \in (f(0),f(\pi))$ so by the Intermediate Value Theorem $\displaystyle \exists c \in (0, \pi) : f(c) = 0$

3. Honestly, I don't understand what is going on. It could be because of how it's presented but I am not sure between conclusions that have been made. Thanks for the help anyway. The general idea is so intuitive to me that I am having trouble understanding the necessity of the steps involved in proving it. I have always been bad with proofs haha.

4. I probably should considered two cases, first if $\displaystyle f(0) < 0$ and then if $\displaystyle f(0)>0$. In the first case will be $\displaystyle f(\pi)>0$ and in the second $\displaystyle f(\pi)<0$. Note that $\displaystyle f(0) \neq 0$. In both cases we showed that the function changes it sign, so by the Intermediate value Theorem we are done.

The point is to show that $\displaystyle f(\theta)$ satifies the Intermediate Value Theorem hypothesis, and that it changes sign. thus it has, at least one root $\displaystyle ( c \in (0, \pi) : f(c) = 0)$