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Worked example: absolute and relative extrema

Extrema is the general name for maximum and minimum points. This video shows how to identify relative and absolute extrema in the graph of a function.

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  • duskpin ultimate style avatar for user Preslav Mihaylov
    At 1.37 Sal said that the specified point is not a relative maximum. According to the definition for a relative maximum:
    f(a) is rel. maxima when all the x near it are f(a) <= f(x)

    In the example, the specified point lies at a position, where the points left of it are all equal to it and the points right of it are less than it. Therefore, doesn't that make the specified point a rel. maxima?
    (55 votes)
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  • male robot hal style avatar for user josh
    Wait a minute, but what if the map shows a function that has two points that are the same height, but says plot the absolute maximum?
    (10 votes)
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    • female robot grace style avatar for user C C
      That scenario shouldn't happen. There will be one absolute max or min if they ask you to mark it (it can happen in a function, you just won't get it as a question since in that case there isn't an absolute max/min).
      (3 votes)
  • male robot donald style avatar for user Shahzeb Arshad
    What is the difference between absolute maximum and global maximum?
    (4 votes)
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  • marcimus red style avatar for user Isha
    At , Sal says that the point (3,-8) would be a relative maximum point but how is that possible? The function is only till -8. How can we assume that the function will have the greatest value considering the points around it? I hope I made my question clear.
    (6 votes)
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    • male robot hal style avatar for user KEVIN
      Sal told us at the beginning of the video that the domain was closed, that is, it included the end points. The domain is [-8,6]. On this particular graph, if we start at x=-8, and move towards the right on the x axis, the next immediate f(x) is less than it was at x=-8. Because we are on a closed interval, that makes the point (-8, 3) a relative maximum. (Make sure you put your x coordinate first when referring to a point on a graph😊.)
      (4 votes)
  • piceratops seed style avatar for user darthwow69
    So what is the difference between absolute max/min point and global max/min point?
    (6 votes)
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    • old spice man green style avatar for user Shin Andrei
      They're the same. Maybe you mean relative max/min point and global/absolute max/min point.
      A relative max/min point is a point higher or lower than the points on both of its sides while a global max/min point is a point that is highest or lowest point in the graph. In other words, there can be multiple relative max/min points while there can only be one global/absolute max/min point.
      (0 votes)
  • leaf green style avatar for user sst
    Is it possible for a function to have multiple global minima? For example, a sin or cos wave has similar value in their minimum/maximum. How those minima and maxima should be called? Do sin wave and cos wave have global minima or do they have only local minima?
    (4 votes)
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    • orange juice squid orange style avatar for user graciousartist
      By definition of absolute/global minimum and maximum you cannot have multiple of these points. You can have multiple points that are the absolute/global min or max though there would still be only 1 absolute/global min or max.

      For example, on the last graph that Sal uses the absolute/global max point is 7. We can have another point on the graph that is 7, but that doesn't mean we have multiple absolute/global max it just means that there are 2 points with the absolute/global max. If another point was created on that was y=8 then 7 would no longer be the absolute/global max because 8 would be the absolute/global max.

      There might be some terminology for this, but I don't know what it is.
      (1 vote)
  • leafers seed style avatar for user Mercades Kenz (-;
    Just making sure, there can be more than one relative minimum or maximum point?
    (3 votes)
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  • spunky sam blue style avatar for user 💎Chυcκ Lørrε💎
    Hey how to get the minimum and maximum of a function without drawing it?
    (3 votes)
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  • purple pi purple style avatar for user louisaandgreta
    So all these max and min, whether relative or absolute, only deal with the range (i.e.: the y axis) not the domain (i.e.: the x axis)? Is that correct?
    And is it always that way?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      You are correct, max is highest it goes and min is lowest it goes, so this matches range (RBT) as bottom to top. Domain (DLR) is left to right, so not related to highest and lowest. That is why most if not all Algebra I functions will have a domain of all real numbers - Algebra II gets into functions such as square root or rational functions which has limited domains.
      (3 votes)
  • blobby green style avatar for user Adam Zerner
    Shouldn't the range, h, be specified?
    (3 votes)
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Video transcript

- [Instructor] 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. So there's two types of relative extrema. You have your relative maximum points, and you have your relative minimum points. And a relative maximum point or relative minimum, they're relatively easy (laughing) 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, and once again since we're talking about the relative maximum, this mountain peak doesn't have to be the highest mountain peak. There could be higher mountains, and actually each of these peaks, 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. So first the relative maximum points. Well that's a top of a hill right over there, this is the top of a hill. You might be tempted to look at that point and 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 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. So 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. So you have an absolute maximum point at let's say x equals c if and only if, so I'll write iff 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, iff, 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, 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 be actually one of, it would happen at one of the endpoints of our domain. So that is our absolute max, and this right over here is our absolute, absolute min. Now once again there is an edge case that you will not see too frequently. So for example, if this function did something like this, so 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 are 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. And so in most problems, it's pretty easy to pick out. Because the absolute highest point on the curve will often be your absolute maximum, and the absolute lowest point on your curve will be your absolute minimum.