We need to start with a definition, and we can start with contrasting what we want the difference between a local maximum/minimum and a global maximum/minimum. Sometimes these are also called absolute maximum/minimums.
If we focus on the terms or the names we’re giving, then the difference should be based on the distinction between the words “local” and “global.” In one, we’re considering some confined and relatively arbitrary geographic area, just the things around or in the neighborhood. In the other, our context grows until we’re considering the whole picture, the whole space that we’re interested in!
Note that the difference between this definition and Definition 4.2.3 Local Maximum/Minimum is the types of \(y\)-values we’re comparing \(f(c)\) to: in this new definition, we just use all of the \(x\)-values in the domain. In the definition for a local max/min, we had to construct some interval to intersect with the domain in order to just consider the “local” picture.
Activity4.4.1.When Would We Not Have Maximums or Minimums?
In this section, we’re going to define these global maximums and then, most importantly, try to predict when these global maximums or global minimums might actually exist for a function.
Come up with some examples of graphs of functions that are bounded (do not ever have \(y\)-values that tend towards infinity in a limit) that do not have some combination of global maximum/minimums.
For the examples of graphs that you have built or collected, what are the features of the functions that allow for the examples you picked? If you could impose some requirements that would “fix” the examples you found (so that they had both a global maximum and a global minimum), what requirements could you use?
SubsectionWhen Do We Guarantee Both a Global Maximum and a Global Minimum?
Theorem4.4.2.Extreme Value Theorem.
If \(f(x)\) is a continuous function on a closed interval \([a,b]\text{,}\) then \(f\) must have both a global maximum and a global minimum on the interval.
The Extreme Value Theorem guarantees the existence of both the global maximums and minimums, but we actually get more than just this out of the Extreme Value Theorem. Once we know that both of the global maximums and minimums exist, we can find them pretty easily.
The global maximum of a function, if it exists for the function on the domain/interval, is just the local maximum with the largest \(y\)-value. Similarly, the global minimum, if it exists, is the local minimum with the lowest \(y\)-value.
So once we know they both exist for a function on an interval, we can simply collect the critical points of the function (including the ending points of the domain) and compare the \(y\)-value function outputs!
Check to see if each function (on the stated domain) satisfies the conditions of the Extreme Value Theorem, and then find any global maximums/minimums of the function on the interval.
SubsectionWhat about Domains of Functions that Aren’t Closed?
Without the conditions that imply the Extreme Value Theorem, things become trickier. For instance, if the function is not continuous, then the function might have some unbounded behavior at a vertical asymptote. In this case, we might need to look at the one-sided limits around that asymptote, in order to see if our function tends towards positive or negative infinity on either side of the asymptote. This could tell us that the function doesn’t have a global maximum, a global minimum, or that it doesn’t have either.
Similarly, if the function is not defined on a closed interval, then we need to investigate what happens to the function’s behavior as the function moves towards the “ends” of the interval. This could mean that the function approaches some real number, but it could also mean that the function approaches positive or negative infinity, or something else. These end behavior limits could exist, in which case we need to compare these heights of horizontal asymptotes or open ends of an interval to the heights of any critical numbers.
All in all, it should be evident that if we remove one or both of the conditions on our function that guarantees the existence of a global maximum and a global minimum, it becomes much harder to find them, since we have so many different options to consider.
To simplify things, we will look at one case where things align in our favor: a continuous function that only has a single local maximum/minimum on an interval.
If \(f(x)\) is a continuous function on some interval, and \(f\) has only a single critical point (call it \((c,f(c))\)) where the direction changes, then if that point is a local maximum of \(f\text{,}\) then \(f(c)\) is the global maximum. Similarly, if \((c,f(c))\) is a local minimum, then \(f(c)\) is the global minimum of \(f\text{.}\)
This is a great result to give us a path forward without having to check all of the edge cases and possibilities mentioned above. There are many functions that might have only a single critical point, or if it does have more than one critical point, only a single one of them acting as a local maximum/minimum.
For each function, describe whether or not the Extreme Value Theorem applies (and explain why or why not). If it does apply, find the absolute maximum and absolute minimum.
