Intro to Extrema - Online Tutor, Practice Problems & Exam Prep

Finding Extrema Graphically

In the past, we've been able to identify extrema on a graph. That's our maximum and minimum values that we know occur at the highest and lowest points of a function, but there's actually a bit more to it than that because there are 2 different types of extrema. There's global extrema, that's our highest and lowest points of our whole entire function, but also local extrema, which are going to be our highest and lowest points when considering just part of a function. Now here I'm going to walk you through the difference between these two types of extrema and also show you exactly how to identify them on a graph. So let's go ahead and get started.

Now in our example here, we are asked to determine whether each labeled point is a global max or min, a local max or min, or none of these at all. So let's start here by talking about global extrema. Now, you may also hear global extrema referred to as being absolute extrema, but just know that this means the same exact thing. Remember that earlier I said that global extrema meant we were going to consider our whole entire function. So let's get a bit more specific here.

Now we're going to be working with this x value c here. And if we plug c into our function, so we get f(c), if f(c) is then greater than or equal to f(x) for all other x values of our function, that means that this point represents a global maximum. Now on a graph, that just means that if we take a look at our whole entire function, our highest point of that whole function is our global maximum. So looking at this function specifically, I can see that my highest point is right here. So this point is my global maximum.

Now there are a couple of different ways that I could state this as an answer. I could express this as an ordered pair and say that my global maximum of my function is located at negative 35. Or I could also say that my function reaches a global maximum value of 5 at x equals negative 3. Both of these would be correct answers. Now that was our global maximum, but let's look at our global minimum.

If f(c) is instead less than or equal to f(x) for all x, that point is then a global minimum. Looking at a graph, remember, if we look at the whole entire function because we're working with global extreme values here, for a global minimum, this is going to be our lowest point of our entire function. And for this function specifically, I can see that that happens right here at negative 5, negative 5. So this point is my global minimum. So for our global extrema, we want to look at our whole entire function and identify our highest and lowest points.

Now let's shift gears to our local extrema. Now, you may also hear local extrema referred to as a relative extrema. But again, this means the same exact thing. Now for global extrema, we are looking at the whole function. For local extrema, we're now just looking at part of it.

But what exactly do I mean by that? Well, if we're looking at a local maximum, then f(c) is going to be greater than or equal to f(x) but for all nearby x. So instead of for all x like for our global maximum, this is just for all x nearby because it's just a local maximum. Now what does this look like? Well, coming over to our graph here, if I were to zoom in on this little region of my graph, I can see that this point at 24, we are higher than all of those nearby points.

So this point is a local maximum. Now what about our local minimum? Well, for our local minimum, this is going to be where f(c) is less than or equal to f(x), again, just for all nearby x values. So again, coming over here to our graph, if I zoom in on this little region here, I can see that we formed this little valley where this point 1 is. And it is lower than all of those nearby points on my graph.

So that means that this point is a local minimum. So our local extrema are just our highest and lowest points within a small region. These are typically going to look like the peaks and valleys that we see on our graph here. Now we've looked at both global and local extrema, but some points may actually be both a global value and a local value. So what does that look like?

Well, if we come back over here to our global maximum and if we zoom in just on that particular region and just look directly around it, we can see that this point, negative 35, is higher than all of those nearby points. That means that this point is not only a global maximum but also a local maximum because it's not only the highest point of our entire function, but it's also the highest point in just this particular region. Now if our global maximum is also a local maximum, what about our global minimum? Well, since our global minimum is actually the endpoint of our function here, things are going to get a bit more complicated. Now this is one of those things that in math not everyone agrees on, so we have to choose a convention to use.

Now by the convention that we're going to use here in this course, endpoints can be global extrema. Like here, this endpoint is our global minimum. But they cannot be local extrema, meaning that this endpoint cannot also then be a local minimum as well as a global because it's an endpoint. And when looking at our other endpoint here, we know that this is not a global max or min because it's not the highest or lowest value of our entire function, but it also cannot be a local max or min specifically because it's an endpoint. Now remember that this is something that can vary between your textbook or professor, so be sure to verify what convention you should use in your coursework specifically.

Now that we know what global and local extrema are and how to identify them on a graph, let's get more practice together. I'll see you in the next one.

Finding Extrema Graphically Example 1

In this problem, we're asked to identify the extreme values of the function f(x) on the specified domain. Now, this is actually the same exact function in all of these graphs. The difference is the domain over which this function is defined. So let's go ahead and start all the way over on the left here with our first graph. Now looking at this function, it is defined over the domain of all real numbers from negative infinity to positive infinity.

So, what extreme values, what global maxima, and minima or local maxima, and minima does this function have? Well, looking at this function, I can see that the absolute lowest point on my graph is down here at x equals 0. Now, because this is the lowest point on my function, I can say that this is a global minimum. Now also, because this point is not an endpoint and I can kind of zoom in here, I can see that this is the lowest point compared to all nearby points. So, this is not only a global minimum, but it is also a local minimum.

So, we have a global and local minimum of 0 at x equals 0. This is our global and local minimum point. Now, when thinking about maximum values, looking at this function, since it's defined from negative infinity to positive infinity, I can see that my graph continues going off to infinity on both sides of this function, meaning that there's not one single maximum value that this function reaches. So, there are no maximum values for this function. There's only a minimum.

Let's move on to our next graph here. Here, we have the same function but now defined over the domain from 0 to 3, with both of those endpoints, 0 and 3, included in that domain. So, what are our maximum and minimum values here? Well, I can see that my very lowest point on my function is here at x equals 0. So, this point represents my global minimum.

Now, you may be thinking here, is this also a local minimum? But remember, because of the convention that we use in this course, endpoints cannot be local extrema. So, this point is only a global minimum. So, we have our global minimum of 0 at x equals 0. Now remember, this is something that may vary between your textbook or professor.

So, again, be sure to verify what convention you should use when it comes to considering endpoints as local extrema. Now, what about the other side of our function here? I can see that this very highest point happens here at x equals 3. So, this is again an endpoint. It can't be a local maximum, but it can be my global maximum, and it is because it's that very highest point. So, this function has a global maximum of 9, and it reaches that point at x equals 3.

So, this function has both a global minimum and a global maximum, but no local extrema because those endpoints were my global min and max. Now, let's move on to our last function here. Here, we see that our domain is defined from 0 to 3, but this time, my endpoints are not included in that domain. So, what are the extreme values of this function? Well, when we think about reaching the very lowest point, that would be our global minimum, we would think about this point here at 0.

But, this point is not included in our domain, so we can't say that that's where our global minimum is. Now, you might be thinking, well, couldn't our global minimum be at 0.01 or 0.001? That's exactly the problem that we're encountering here. We can get closer and closer to 0, but we can never quite reach it.

And there's always going to be a value on our function that is smaller than whatever we're thinking of. Even if I said 0.0001, you could still get smaller. So, there are no local or global extrema on this function because we come to the same conclusion when looking at this other point. We could say, okay, we could get to 2.99 or 2.999 to get to that maximum value.

But, remember that you can always get a higher value because we can't ever reach that endpoint because it's not included in our domain. So, for this last function here, there are no extreme values that we can identify. So, looking at these functions, we can see that even though they're all the same exact function, because they're defined over different domains, our extreme values look completely different. Now, be sure to keep this in mind as we continue practicing. I'll see you in the next one.

Finding Extrema Graphically Example 2

In this problem, we're asked to label where the local and global maxima and minima occur on the graph of this function f of x. So let's go ahead and dive in. Now looking at our function here, remember that maxes and mins are most often going to occur where there are these sort of peaks and valleys that we see happen. So let's go ahead and take a look at a couple of these points. Now looking at my function as a whole, I can see that the very highest point of this entire function is right here at x equals negative 2.

Now because this is the highest point of my entire function, that tells me that this is my global maximum. But, again, if I were to zoom in on this point, I can see that it's also the highest point of those points nearby. So this is not only a global maximum but also a local maximum. Now this will pretty much always happen unless your global max or min is at an endpoint. Okay.

So let's keep going here. Now that was our highest point of our entire function. We also want to take a look at our very lowest point of our function, which I can see happens here at negative 5 when x is equal to 1. So that tells me that this point is my global minimum. And remember, if I were to zoom in here, it's also the lowest point of all those nearby.

So this is both a global and a local minimum. Okay. Now we want to find if there are any other local maxes and mins. There can't be any other global because we already have our absolute highest and lowest points, but there can, of course, be more local points. I want to take a look at all of these little peaks and valleys that happen.

So over here at x equals negative 4, I can see this little valley forming. And if I were to zoom into this point here, I can see that it is the lowest point of all of those nearby points. So this is a local minimum. Now shifting over to this side of my graph over here, I can see that my function's graph is reaching a sort of peak here at x equals positive 3. So that tells me that zooming in here, I can see that this is the highest point of all those nearby points.

This point is a local maximum. Now I see this valley forming down here at x equals 5, again, cluing me in that this is a local minimum. If I zoom into just that point, I can see it's lower than all of my nearby points. This point is a local minimum. Now something you may be wondering here is, what about those endpoints?

Remember that based on the convention we will use in this course, endpoints cannot be local extrema. And because these are not global extrema or endpoints, they're actually not extreme values at all. So here, we've identified all of our local and global maxima and minima. Feel free to let me know if you have questions, and let's keep practicing.

Finding Extrema Graphically Example 3

In this problem, we're asked to determine where the local and global maxima and minima occur on the graph of our function f(x). We're given the graph of our sine function. At first glance, this might seem a bit tricky because there may seem to be many maxima and minima, since we know that the sine function oscillates up and down between 1 and negative one. Let's go ahead and dive in here.

Because we've been working with functions that just have one highest point and one lowest point, this may seem confusing. However, because all of these peaks and valleys occur at the same exact high point and the same exact low point, we actually have multiple points representing global maxima or global minima. Looking at this highest point at \( x = \frac{\pi}{2} \), we know that it reaches a value of 1. This is the highest point that this function ever reaches, indicating that this point is a global maximum.

This point is also a local maximum because it is higher than all nearby points, and it is not at the endpoint. Thus, it is both a global and local maximum. Now, looking over here at \( x = \frac{5\pi}{2} \), you might wonder whether this point, which also reaches 1, is also a global max. The answer is yes. Because the absolute highest value that this function reaches is 1, it's acceptable for it to achieve this value multiple times. This point, too, is both a global and local maximum.

The same principle applies to our minimum values. We know the absolute minimum value that the sine function reaches is negative one. Therefore, at \( x = \frac{3\pi}{2} \) and \( x = \frac{7\pi}{2} \), both points represent both a global and local minimum value. Dealing with these oscillating functions can be confusing, but it's understandable for a function to have multiple global maxima or minima since it reaches those values multiple times. It remains the highest or lowest point of the entire function. Feel free to let us know if you have any questions.

Let's keep going.

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

• A. Absolute max of 4 at $x=-2$, Absolute min of $-3$ at $x=5$
• B. Absolute max of 4 at $x=-2$, Local max of 4 at $x=-2$, No absolute or local mins
• C. Absolute max of 4 at $x=-2$, Absolute min of 2 at $x=-5$
• D. Absolute max of 4 at $x=-2$, Local max of 4 at $x=-2$, Local min of 2 at $x=-5$

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

• A. Absolute max of 5 at $x=4$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$ & 1.5 at $x=-4.5$
• B. Absolute max of 5 at $x=4$, Absolute min of $-2$ at $x=1$, No local extrema
• C. Absolute max of 4 at $x=-3$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$
• D. Absolute max of 5 at $x=4$, Local max of 4 at $x=-3$, Absolute min of $-2$ at $x=1$, Local min of $-2$ at $x=1$

Finding Extrema Graphically Example 4

If you haven't already, go ahead and try to solve this problem on your own before checking back in with me. Let's go ahead and dive in here. Now here, we're asked to sketch a graph of a function that is continuous over the interval from negative 3 to 2, both of those endpoints included, and it has the properties that are given here below: an absolute maximum at x equals 2, an absolute minimum at x equals 0, and a local maximum at x equals negative 2. Now keep in mind that if you already tried this on your own, your function might look a little bit different than mine because we can still have different things happening as long as we have all of these absolute and local maxes and mins, and our endpoints are the same. So let's go ahead and start sketching our graph here.

I know that I have to have endpoints at negative 3 to 2, and I know that those endpoints will be solid dots because those points are included in that interval. So let's think about what needs to happen here. Well, I'm told that I have an absolute maximum at x equals 2, and that's also going to be one of my endpoints. So I know that this needs to be the highest value that my function reaches because it's an absolute max. So I'm going to go ahead and just pick a higher value up here.

Now, keep in mind that your absolute maximum could actually be some value that's down here under your x-axis as long as it's still the highest point of your entire function. I'm gonna go ahead and keep my absolute maximum somewhere up here. It doesn't matter where yours is as long as it's the highest point of your entire function. So that's my absolute max. Then I have an absolute minimum that's the lowest point of my entire function at x equals 0.

So I'm gonna go ahead and put that point somewhere down here on my function. And I know that because this is not an endpoint, it's going to end up looking like some sort of valley or, otherwise, some sort of downward peak that we're coming to. Now, finally, we have a local maximum at x equals negative 2. So that means that I need to reach some sort of peak there. So let's go ahead and mark a point at x equals negative 2.

Now remember, this is just a local maximum, so I need to make sure that this does not go above where my absolute maximum is. So I'm gonna go ahead and just put that right here. I know that it needs to look like some sort of peak. Doesn't matter what exactly that peak looks like. Okay.

So my final thing is that I need to include that endpoint negative 3, and that's not any extreme value because remember, endpoints cannot be local extrema. And we already have marked our absolute max and min. So So I'm gonna go ahead and just mark that negative 3 right here on my x-axis. That is my other endpoint. So let's go ahead and connect our function.

Remember that this is continuous, so there should be no breaks or holes. So I know that this is an absolute max. I'm gonna go ahead and connect up here and then go straight down to my function where I reach some sort of valley down there. And then finally, I go up to that absolute maximum, and here is my function. Now let's verify that it has everything it needs to.

I have my endpoints at negative 3 and positive 2 based on that interval that was given to me. I do have an absolute maximum at x equals 2. That's the very highest point of my function. I have my absolute minimum down here at x equals 0. That's my lowest point of the function.

And then finally, I have a local max at x equals negative 2. It reaches a peak. It's higher than all those nearby points, but it's not the highest point of my entire function. So I've successfully sketched my graph here. Let us know if you have questions, and let's keep working.