What are the applications of the extreme value theorem?

Question

I recently started studying about global maxima and minima.

The extreme value theorem proves the existence of global maxima and minima for a continuous function in a closed interval.This theorem is actually, pretty much intuitive and looks like a basic property of a continuous function.

Has it got any applications other than just proving existence of global maxima or minima?

If you take a look at Lagrange's mean value theorem we can use it to prove inequalities like $|\cos(a)-\cos (b)|<|a-b|$.

So, I'm just looking for some cool stuff that you can do using the extreme value theorem.

Answer 1

The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.

LUB Property $\rightarrow$ Monotone bounded convergence $\rightarrow$ Bolzano-Weierstrass $\rightarrow$ EVT $\rightarrow$ IVT $\rightarrow$ Rolle's Theorem $\rightarrow$ MVT, Cauchy's MVT $\rightarrow$ Integral MVT

Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.

Answer 2

Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f \in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = \frac{1}{x}$, then it's continuous on $(0,1)$, but not bounded.

Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$

Indeed, by MVT, for any $x,y\in (a,b)$, we can find a $\zeta \in (a,b)$ such that \begin{equation} |f(x)-f(y)| = |f'(\zeta)||x-y| \leq M|x-y|\end{equation} so we see $f$ is not just uniformly continuous, but Lipschitz.

Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m \leq f \leq M$, where $f:[0,1] \rightarrow \mathbb{R}$. Then we have \begin{equation} \int_{0}^{1} m \leq \int_{0}^{1} f \leq \int_{0}^{1} M\end{equation} Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $\zeta$ such that $f(\zeta) = \int_{0}^{1} f$

In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.