Extreme Value Theorem

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We develop the Extreme Value Theorem, a way to know when an absolute extrema must exist without doing calculus.

Video Lecture

Text and Additional Details

As we continue we will encounter many circumstances where it would be helpful to know if there is a highest or lowest attained value for a function, i.e. absolute extrema. This can be a surprisingly difficult thing to determine in some cases, but in others we can know quite quickly whether or not we can expect to find such points - this will lead us to our second existence theorem, the Extreme Value Theorem (EVT).

Before we can tackle the extreme value theorem though, we need to introduce the idea of a compact set. The actual mathematical definition of a compact set is very complex (which is why looking it up on something like wikipedia will probably make your brain explode or just look like total gibberish) but for our purposes we can boil it down to the following two properties:

Compact Set A set is said to be compact if it is:

Bounded, i.e. there is some number (not necessarily in the set, but an actual number, not $\infty$ ), called an ’upper bound’, that is larger (further right on the number line) than any number in the set, and another number (as before, not necessarily in the set but not $-\infty$ ), called a ’lower bound’, that is smaller (further left on the number line, i.e. more negative) than any number in the set.
Closed, i.e. for every interval in the set, the endpoints of the interval are also in the set.

Again, the technical mathematical definition of bounded and closed are a little more complicated than what is listed here, but the definition given is sufficient for the calculus sequence. As usual this is easier to understand using some examples:

$\left \lbrace x : x \leq 5 \right \rbrace$	This is the same as $(-\infty ,5]$ and is not compact since it has no lower bound.
$\left \lbrace x : 3 \leq x \leq 7 \right \rbrace$	This is equivalent to $[3,7]$ and is compact.
	since $2$ is smaller than any number in the set,
	and $8$ is larger than any number in the set (so it is bounded),
	and both endpoints, e.g. $3$ , $7$ , are included (so it is closed).
$(-22, 12]$	This is not compact because it is not closed, since it is missing the endpoint $-22$ .
$[-3,17] \cup [18, 25]$	This is compact.
	The value $-10$ is lower than any value in the set,
	and $30$ is larger than any number in the set, so it is bounded.
	It also includes all the endpoints, e.g. $-3$ , $17$ , $18$ , and $25$ , so it is closed.
$[-12, -1] \cup [0, 5)$	This is not compact since it is missing the endpoint $5$ .

Which of the following are compact sets (select all that apply)

$(-\infty , \infty )$ $[-3, 5)$ $[-3, 5]$ $[-3, 5) \cup (7,13]$ $[-3, 5)\cup (6,13]$ $[-5, 3] \cup [10,13]$ $(-12, 17) \cup [22,\infty )$ $[-31, -15] \cup [1,\infty )$ $[-31415, -\pi ] \cup [e,27]$ $[-\pi , e]$ $[12, 777]$ $(-67, 31] \cup (33,35] \cup [40,41]$

Now that have a handle on what a compact set is, we can state the extreme value theorem.

Extreme Value Theorem (EVT) Let $f:\mathbb {R}\rightarrow \mathbb {R}$ be a function that is continuous everywhere on the compact set $D\subset \mathbb {R}$ . Then $f(x)$ has both an absolute maximum and an absolute minimum value on $D$ .

So, in human speak, this means that if you have a function and some compact set, like a closed interval $[a,b]$ , as long as $f(x)$ is continuous on that compact set, then you know $f(x)$ has an absolute maximum and an absolute minimum somewhere on that compact set. It’s important to keep in mind that this isn’t necessarily the highest or lowest point that $f$ can hit overall; just on that compact set we were considering.

Just like the intermediate value theorem, the extreme value theorem is an existence theorem. It doesn’t tell us where the function attains its highest or lowest points, rather it only tells us that it has a highest and lowest value.

To see why we know the function must attain it’s maximum and minimum, watch the above video for a nice visual explanation.

So we learned what a compact set was; in particular a set that is bounded above and below, and includes all the endpoints of each segment it contains. Once we had that, we could state the extreme value theorem, which tells us that absolute extrema of continuous functions exist on a compact set. We will see in future videos why this is useful, but it turns out to be pretty helpful to know there are absolute extrema before trying to look for them.

EVT is an existence theorem which means...

It tells you that something exists, but doesn’t tell you where to find it or what it is. The theorem has been proven, hence it exists. The theorem requires something exists before it can be applied. It’s going to give me a headache on an exam at some point.