Member-only story
In this article, we will be introduced to the concept of the Mean Value Theorem. We will see both its definition and how it can be used in practice.
Prelude to the Mean Value Theorem
Before we dive into the Mean Value Theorem, we should first consider another theorem first. This is due to the fact that it will be used later, and this is Rolle’s Theorem. It is defined like so:
Let us now take a look at what this theorem can be used for, by using an example. Imagine that we have the following function:
We now want to prove that this function only has 1 real root, i.e., the rest of the roots are complex. We already know that since our function is a 5th-degree polynomial, then it has 5 roots. However, we need to show only one is real. How can we do that?
We can start by showing it has at least one real root. We can note that f(0) = -2 and f(1) = 10. We can then see that f(0) < 0 < f(1). Now, since f(x) is a polynomial then it’s continuous everywhere. We can also use the Intermediate Value Theorem. We then know that we have a number c, such that we have 0 < c < 1 and f(c)=0. In other words, f(x) has at least one real root.
We then need to show that the four other roots are not real, but instead complex. We will need a contradiction proof. So, let us assume that f(x) has a minimum of two real roots. This means that we can find real numbers a and b such that f(a) = f(b) = 0. We then also know from…
