How to recognize relative and absolute maxima and minima | Functions | Algebra I | Khan Academy

We're asked to mark all the relative extremum points in the graph below. So pause the video and see if you can have a go at that. Just try to maybe look at the screen and in your head see if you can identify the relative extrema.

So now let's do this together. There are two types of relative extrema: you have your relative maximum points, and you have your relative minimum points. A relative maximum point or a relative minimum, they're relatively easy to spot out visually. You will see a relative maximum point as the high point on a hill, and the hill itself doesn't even have to be the highest hill. For example, the curve could go at other parts of the domain of the function could go to higher values. It could also look like the peak of a mountain. Once again, since we're talking about the relative maxima, this mountain peak doesn't have to be the highest mountain peak. There could be higher mountains, and actually, each of these peaks would be a relative maximum point.

Now, relative minima are the opposite; they would be the bottom of your valleys. So that's a relative minimum point. This right over here is a relative minimum point, even if there are other parts of the function that are lower. Now, there's also an edge case for both relative maxima and relative minima, and that's where the graph is flat. So if you have parts of your function where it's just constant, these points would actually be both. For example, if this is our x-axis right over here, that's our x-axis. If this is our y-axis right over there, and if this is x equals c, if you construct an open interval around c, you notice that the value of our function at c, f of c, is at least as large as the values of the function around it, and it is also at least as small as the values of the function around it. So this point would also be considered a relative minimum point.

But that's an edge case that you won't encounter as often. So with that primer out of the way, let's identify the relative extrema. First, the relative maximum points—well, that's the top of a hill right over there; this is the top of a hill. You might be tempted to look at that point at that point, but notice at this point right over here, if you go to the right, you have values that are higher than it, so it's really not at the top of a hill. And right over here, if you go to the left, you have values that are higher than it, so it's also not the top of a hill.

And what about the relative minimum points? Well, this one right over here is a relative minimum point; this one right over here is a relative minimum point, and this one over here is a relative minimum point. Now, let's do an example dealing with absolute extrema. Here we're told to mark the absolute maximum and the absolute minimum points in the graph below. So once again, pause this video and see if you can have a go at this.

You have an absolute maximum point at, let's say, x equals c if and only if. So I'll write "if" for "if and only if" f of c is greater than or equal to f of x for all the x's in the domain of the function. And you have an absolute minimum at x equals c if and only if f of c is less than or equal to f of x for all the x's over the domain. So another way to think about it is the absolute maximum point is the high point. So over here, that is the absolute maximum point. And then the absolute minimum point is interesting because, in this case, it would actually happen at one of the endpoints at our domain. So that is our absolute max, and this right over here is our absolute min.

Now, once again, there is an edge case that you will not see too frequently. For example, if this function did something like this: if it went up like this and then it just stayed flat like this, then this would no longer be an absolute maximum point. But any of these points in this flat region, because they're at least as high as any other points on our entire curve, any of those could be considered absolute maximum points. But we aren't dealing with that edge case in this example, and you're less likely to see that. So in most problems, it's pretty easy to pick out because the absolute highest point of the curve will often be your absolute maximum, and the absolute lowest point on your curve will be your absolute minimum.