The Rolle Theorem

Suppose that you have a function, continuous over a closed interval , differentiable over at least this open interval . If this function assumes the same value at two points, then somewhere between those two points exists at least one critical point, a point where the first derivative is zero.

Suppose you have , over the interval
It's continuous and differentiable over it. We're not gonna waste time proving it, you should know that already.

Now,

This function has the same value at two points. What can we learn from this?
We know that the cosine is continuous over the entire real line.

Now by continuity, it has to be like a "wire". Everything connected.
Now since it's zero at both these points, it must go up and come down eventually, or go down and come up eventually, or stay the same between these points. So, there must be a critical point.

Well, is there such point where the derivative of cos(x) is 0 between ?

Yup! Precisely at x = 0, sin(x) = 0. There indeed exists such point.

Closely related to this theorem is another one called Bolzano's Theorem of Existence of Zeroes