Use the graph to state the absolute and local maximum and minimum values of the function.
Absolute maximum value is $f(4)=5 ;$ there is no absolute minimum value; local maximum values are $f(4)=5$ and
$f(6)=4 ;$ local minimum values are $f(2)=2$ and $f(1)=f(5)=3$
we want to find the absolute and local maximum and minimum values of the given function. In this case of function it's given through the graph like this. This is a sketch of the graph given in the text book but we have preserve the essential properties of the function. So we had those uh huh opens red circles and pull those like I'm talking about this one here, it is an open red red circle. This an open red circle. It's an open research that means that the points corresponding there are not included in the graph but when we have a feel red circle like this then we know Like this one I'm talking about that the point correspondent point is inside correspond to the graft as a way of saying that graphically we have those going there and without we want to find the log the absolute extreme math extreme of the function and the local maximum minimum. So let's start it's always with the absolute streams which are essentially yeah, is to find. So let's talk about the absolute maximum value and when we talk about that we are saying at the same time at which point that absolute maximum of course. And in this case we get to look at the entire domain In this case we said that the maintenance the interval from zero closed in syria because syria has emerged to because they start here, this circle is filled with red. So meaning that the point is part of the graph. So the image your is too that's called this function F. Haven't said it's called F. So um The domain goes from zero All the way through seven, but 7 is open. So we came right here to the main of F is the interval from zero close at zero but open at seven. Uh And that's why because the red circle is not filled with red. That is open, meaning that the point is not included in the graph. That's very important. That makes a huge difference. Okay, back in all the points between Syria and seven, including Syria only uh have an image for example, the image of 123. The thing is that it is extracted or separated from the line that has a train trend here. This red line has the trend coming from the left and then continued from the right but that specifically point specifically point. Uh one the image has been instructed we were expecting for but it was um got out of the trend of the line and put at high three. That is the image of 13 Now in this graph. And what happened at six, he said the from the left graph stopped here including the point and continues then at this point, but not including the point because number six get to have only one image can have two images because in that case we won't have a function. So this is a situation here a little bit. And then with that in mind We look at the whole domain that is from 0-7, including Syria and not seven. Um to uh find the absolute maximum value and in this case would be the highest value or the largest value of f possible. In this case we can see that that happens at four. And the values five correspondent to in the draft at this point here. So the absolute maximum value is yeah, if at four which is equal to five. So the accident actually, maximum, body is five and it's attained or course at x equals four. Now, let's talk about the absolute minimum in this case, respected these to be this, as the graphic suggest, is the lowest value of the lowest point in the grass. That is the smallest value of the function. But the problem here is that the point here is not included, seven is not in the domain. So, uh we can say that the function decreases all the time getting near seven but never reaches an image at that point. And for that reason, there is no absolute minimum value for this function. So this absolute minimum does not exist. And that's very important to understand the only way that this absolute minimum B one at 7 is that we include the point to the graph, that is if we do something like this, including the point, but because we don't have that, that is t point is excluded explicitly by not feeling the circle, we can say that there is no lowest possible value to the graph or equivalently absolute minimum value for this function does not exist. Let's talk about local for example local maximum. So local maximum at the dysfunction is for example at four. It is at four this point here. Is that the global or absolute maximum we saw here, but it's also local Maximum because if we look close or near the value four here, that is, we look only a portion of the graph. Years, the function we see that in effect again, the highest value of the largest values of function is five attained at four. So before we have a local maximum But there is another one Let's say at at six we have again uh local maximum If at six and the image of six is 4 because that value is included in the graph. Because we have the field circle here. Just So the image of six is four and that's a local uh let's see this fight. Sorry to remarked upon in green And if you look close to six Oh near six, we consider that the largest image Or value of the function is effectively four at 6. So it was the local maximum. Okay, and there are no more local maximum because one happens. Another thing that we're going to talk right now. So first let's go to the local minimum now local minimum and we can see that the local minimum, let's put it in blue, we have one here because if we look around five close to five, we can see that this modest value of the function is the image at five precisely. Which is three. So there's a local minimum if At five equal 3. But there is another one. Um here, let's say these founders put it in blue around this point. There is local minimum. If you look close to value to we have local minimum. That case we got to talk about. Mm hmm. But we need your mouth. Okay, let's put it uh huh. Back here and above was also there are two values and we can talk about, oh that's me At four FF 40 564. And local minimum is 5.3. Yeah, I'm the one and 2 if I had to equal equal to and add one, The image of 1, 3. Because the point is uh huh Good. A party trend line that is this point here is not including the graph of the point is who has been put here down here. And then You're saying is that we look close to one and in this case Remembrance, it is important to look close to the point that we don't need to go far from the point. We always got to be and we think near the point. So we look close to the number one. We can see that the graph is something like this and the point a separate point at one Who's in that? She's three. So we can say they're looking closely to want the value Effort Wanna Go three is a local minimum. So we got to put it in the local minimum list. So we have ever six f sorry, efforts, five equal three local minimum, same thing for effort to local minimum. And F at one equals 3. So we have three local minimum and two local maximum. And of course we have an absolute maximum value Effort for Article five and there is no actual minimum value of the function Because seven is not included in the domain.