In this video, I will present the Extreme Value Theorem, a very important theorem about continuous functions. Before telling you the full statement, here is what the theorem is about. The theorem tells us that, under certain conditions, a function is guaranteed to have a maximum and a minimum.
I will spend some time trying to justify what these conditions are. To clarify, this theorem does not tell us how to compute the maximum or the minimum. It just tells us that they exist.
Nothing more. So, once again, I am searching for some simple, sufficient conditions that will tell me that a function must have a maximum and a minimum without having to do much work. Before going any further, let's make sure we understand what a maximum and a minimum are. I will write a rigorous definition for maximum.
The definition of minimum is similar. I want to say what it means for a function to have a maximum, like the one in this graph. This function has a maximum at the point z. Having a maximum has something to do with this inequality.
The value f is greater than or equal to other values f. As I wrote it, this cannot be the definition yet. This is meaningless.
I need to specify what I am saying about c and what I am saying about x. It should be clear that it is not enough for the inequality to be satisfied for just one value of x. I want it to be true for all values of x. All values of x where? In the domain.
I have to change this a little bit. I am not just defining f as a maximum, but f as a maximum on a set i, which is the domain. This matters, as I may encounter a function that has a maximum in one set, but not on another. Now I can request the inequality to be true for all values of x in i. What about c?
c must also be in i, and it is just one value. So I need to say that there exists c in i, such that, for all x in i, f is less than or equal to f. And now I got it.
This is the rigorous definition of maximum of a function on a domain. For completeness, let's clarify. If we talk about the maximum of a function, the maximum is the value f, the output.
But we also say that the function has a maximum at c. So the maximum is the value of the output, and it happens at the input. And one more definition.
If we do not specify where, if we ever say just... The maximum of f? Then we mean the maximum of f on its domain, on the largest set that the function is defined. Great. Now we have a definition.
Let's explore some examples. Let's look at some functions that do not have a maximum, and let's try to understand why. This was the example I was looking at.
The function in this graph has a maximum at a point in the interior of the interval. This second example also has a maximum, in this case at an endpoint, which is fine. This third example does not have a maximum. Notice that the endpoint is not included.
The function is bounded, but it does not have a maximum on its domain. And this fourth example does not have a maximum either. In this case, there is a vertical asymptote, and the function isn't even bounded.
It grows arbitrarily large. I notice a pattern in these examples. In the first example, there is a maximum, and the domain is a closed and bounded interval, an interval that includes both endpoints.
The same is true in the second example. The domain is an interval that includes both endpoints, and the function has a maximum. But in this third example, the domain is missing one endpoint. The function does not have a maximum, and the lack of a maximum seems related to the missing endpoint. The same is true in the fourth example.
The lack of a maximum also seems related to the missing endpoint in the domain. However, there are also other types of functions without maxima. For example, this one.
The domain is an interval that includes both endpoints, and yet this function STILL doesn't have a maximum. In this case, the problem is not the domain. It's the function itself.
This function is bad. And by bad, I mean it is not continuous. The discontinuity is what allows the lack of a maximum.
All these examples together suggest that to guarantee a maximum, I need a continuous function on an interval that includes both endpoints. That is exactly what the Extreme Value Theorem says. If f is a continuous function on a closed and bounded interval, an interval a, b, that includes both endpoints, then it must have a maximum and a minimum. This theorem needs a proof, of course, but I will not write it here. Even though the theorem is easy to understand and use, it is quite difficult to prove.
The proof is long and technical. You would normally learn such a proof in a rigorous analysis course, where you construct the reals axiomatically. For our purposes, hopefully the examples in this video have persuaded you that the statement is reasonable, and we can take it just as an action, instead of as a theorem.
To finish the video, I will leave you with a challenge. Can you construct a function that... Is continuous on the interval, missing 0 but including 1? Does not have a maximum, and does not have a minimum either. The examples I presented in a domain like this, missing only one endpoint, were missing a maximum but had a minimum.
Can you construct a function that is missing both? This is not a trick question. It is doable. As a hint, try to sketch the graph of such a function first, and only once you have an idea of the graph, try to come up with an equation.