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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.

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  • $\begingroup$ It's used in the proof of many other theorems. $\endgroup$
    – Dole
    Jul 28, 2018 at 5:24
  • $\begingroup$ By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?) $\endgroup$
    – Bob
    Jul 28, 2018 at 6:04
  • $\begingroup$ Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic). $\endgroup$ Jul 28, 2018 at 6:53

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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.

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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.

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