 In the previous video, we learned about the idea of a local extreme. I'm a local max or local man in this video We're gonna actually talk about a related concept known as the absolute maximum or the absolute minimum or The neutral term here would be absolute Extremum because with a local Maximum all that means is that it's the biggest point in some neighborhood knowing I got bigger than it in that neighborhood But the absolute maximum is referencing the the very biggest Why coordinate in the range of the function and similarly the absolute minimum will describe the very very very smallest Why coordinate in the entire range and then absolute extrema means it's it's either an absolute maximum or absolute minimum And so if you look at this graph right here, for example, we can very quickly identify the local extreme We see that there's a local minimum right here at x equals negative one There's another local minimum right here at x equals three and then finally there is a local maximum right here This is a local maximum at x equals one now if we ask the question about absolute Extremum I want to note that there is an absolute max a minimum right here at x equals three Right. This is not just a local minimum This is the absolute minimum because there is no why coordinate on this graph that ever gets smaller than y equals three The local minimum here at x equals negative one. It's a local minimum But it's not the absolute because we get smaller values. What about the absolute maximum here What would be the biggest y-coordinate in this functions range? Well, we're tempted to say that the local maximum is an absolute maximum, right? But that's actually not the case because it turns out that this y-coordinate of two is not the biggest one It can go up to three or four or five or six or seven or eight or nine or keep on going off the screen, right? There y equals two is not the biggest y-coordinate So this one right here is not going to be the absolute maximum just like this wasn't the absolute minimum But what is the absolute maximum then right? You'll notice that is if we look at the picture right here It seems to be increasing right so the y-coordinate is going to get bigger and bigger and bigger and bigger So if we allow x to go as far as we want right So if we allow x to go to the far far far far far right Then the y-coordinate is going to go up and up and up and up and up and up and up and up and up That's what this arrow here is indicating It's telling us that if we continue to the right the function will continue to get bigger and bigger and bigger and bigger And so to shorthand this we often say things like the following as x approaches infinity, right? As x goes to the far right then y approaches infinity as well. That's what this Northeast arrow represents on the graph Similarly, if we look at this arrowhead over here, this tells us that as x goes to the far left The function will continue to go up and up and up and up So this north western arrowhead is telling us that as x approaches negative infinity Then why will likewise approach infinity as well? So because why is approaching infinity? There is no absolute maximum the absolute that maximum doesn't exist Because the y-coordinate to get bigger and bigger and bigger and therefore there is no biggest number in the range Infinity is itself not a number so we can't really count that one So this is an example of a function which has an absolute minimum but has no absolute maximum Some other examples you could take this parabola for example this parabola has an absolute maximum at It's y intercept It's also a local maximum of course notice of course that every absolute extremum will be a local extremum But local extrema don't have to be absolute extremely we see And also this function much like the one we saw previously. It's absolute minimum Does not exist because the function if you look over here this downward Trajectory this south-western arrow here is telling us that as x goes to infinity Y is going to approach negative infinity and that's right here tells us that there's not an absolute minimum and likewise as we go on this South western south western arrow, right? You're going to get as x approaches negative infinity. Oh boy negative infinity y will likewise approach negative infinity as well And so this function doesn't have an absolute minimum in this example We can see what was actually kind of looks like the first one we're seeing in this video, right? We can see that there's again a local a local max right here At x equals 1 but that is not an absolute maximum again This thing is pointing upwards, right as x approaches plus or minus infinity, right? Look on the other side you're doing the same thing as x approaches plus or minus infinity y approaches infinity And so that tells us that the absolute maximum doesn't exist again Because why is going towards the infinity on the other hand though we have at x equals negative one We have a point which is a local minimum at x equals three We have a point which is also a local minimum and you'll notice that in both situations the y-coordinate is zero So who's the absolute minimum is the absolute minimum x at x equals negative one or at x equals three And the answer is going to be both. It's a tie because both of them obtain the smallest value in the range they're both absolute minima and They're both Absolute the minima because they're their y-coordinates are equal to each other and they are the smallest in the entire range So it is possible to have multiple Absolute extrema, but that's because their y-coordinates actually agree with each other And this is how you can find the absolute extrema on the graph of a function