The Best Math Theorem Ever
The Intermediate Value Theorem (IVT) is definitely the best math theorem ever. The IVT is stated thus:
Suppose f(x) is a continuous function on [a, b]. Furthermore, suppose f(a) is negative and f(b) is positive. Then, there exists c element of [a, b] such that f(c) = 0.
The IVT may be stated generally:
Suppose f(x) is continuous on [a, b]. Then, if d is an element of [f(a), f(b)], there exists c element of [a, b] such that f(c) = d.
This is definitely the best math theorem ever, and you can use it to prove lots of things. For example, using the IVT, you can prove that every polynomial of an odd degree has a real root. If a proof requires the use of the IVT, it is probably a very simple proof. Simple proofs require less time and effort, and that is good for college students.
The Mean Value Theorem (MVT) is almost as cool as the IVT. The MVT is stated thus:
Suppose f(x) is differentiable on [a, b]. Furthermore, suppose
f(a) = f(b). Then, there exists c element of [a, b] such that
f'(c) = 0.
Many people think that the MVT is the best ever. Indeed, it is not, simply because the MVT is proved using the IVT.
The proof that the square root of 2 is irrational is pretty cool, but is too common to be the best ever. Honorable mention goes to anything proved by mathematical induction. Induction is cool.
Also, the proof that e(i * pi) + 1 = 0 is wicked cool, and is only marginally less cool than the IVT.