If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

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.

Want to join the conversation?

• 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?
• I had just watched the previous video and I thought the same. If it was for me to do, I would call this a relative maximum point
• 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?
• 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).
• What is the difference between absolute maximum and global maximum?
• 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.
• 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😊.)
• So what is the difference between absolute max/min point and global max/min point?
• 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.
• 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?
• 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)
• At , Sal marked two points as relative minimum points that looked like absolute minimum points. Was there a reason for doing this?
• If they were not the very last points, and still on a sort of bump or hill, then they can still be relative points.
• I do not understand the scenario he gave at , is this related to the example? how?
• That part was a bit confusing for me too. He's just writing the definition of absolute maximum and minimum points in more formulaic terms. An easy way to say it in English is "the highest part of the line is the absolute maximum." The math-language version makes it clear that you'll have an absolute maximum point at (c, y) if the function's output for the input of c is greater than the outputs you get for any other input within the function.

The f(c) part means "what the function spits out when you put c into it", and it also means a value of y for a coordinate on the graph.

For example, if you had a graph that was a diagonal line going from the origin up to the point (4,8) then the formula would look like this:
Abs. value at x = 4 iff f(4) > f(x)

I hope that helps!