(a) Sketch the graph of a function that has two local maximum, one local minimum, and no absolute minimum.
(b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.
Graphs in the video
we want to find a bar a sketch of a graph of a function that has to local maximum, one local minimum and no absolute minimum barbie. We sketch the graph of functions that have three local minimum. two local maximum and seven critical numbers. Let's start with a that is a function that has to local maximum, one local minimum. And now as the minimum seems to be easy in this case because we have something like this where we have let's Let's see two local maximum. So we have this point here over here is a local maximum and this point over here is the local maximum. So we have to local maximum only in this graph we have then one local minimum and this one is here one local minimum, there is only one and now we want no absolute minimum. And in this case let's see that the lowest point in the graphic which should be the absolute minimum should happen here. But the point has been removed just the image of this value here as fame put this here up and for that reason we uh have a graph of the function that do not does not obtain it, that was value. Yeah. So uh we can say that in this craft there is no absolute minimum, there is some sort of maximum but now as the minimum as we want it. So two local maxima. Ah those points here and here and here one local minimum you at this point sorry and no absolute minimum because we have these behavior here on the lowest point should be here but we have removed the image and put it on, put it above in order to not attain the absolute minimum because uh something similar if we don't want an absolute maximum that is we can't put the highest point on the graph to the right and removed that point of the graph and put it up below. That's uh the similar case or no absolute maximum. So we have these three property on this graph. Okay, now we go for barbie which should be a little bit more interesting. And we see the following, we have three local minimum to local maximum. It seems that there are already there five critical numbers. We need two more to have seven curriculum. So we have thrown this and as we can see here we have Uh let's do it with the statement three local minimum. So we have here local minimum here local minimum here and local minimum here. So we have three local minimum. Mhm And we wanted uh two local maximum and we have them up here and here it could be differential. There is no need that is be like this. But this way, but you can have a perfectly differential function here and here and each of these local maximum. So to local maximum at these two points as we said And now we have these five points here are critical numbers because there we have either There are derivative in which case to be zero because they are local extremes or there are these are points where there is no derivative like this in this example and they are also critical numbers because they are in the domain and function and the derivative that does not exceed that exists at those points. So it means that We have already with this property here. We have five critical numbers. So we need two more and the idea is to make the graphic discontinuity with the chump discontinued like this with this graph, we don't have neither local minimum nor nor local maximum at these two points because if we take any interval around those points, there is always the images are greater than the image of the point and lower the the image of the point. So these two points are now extreme by local stream values but but they are critical numbers because they are in the domain of the function. We have put images to those points in this case, we have put the image up here and here at this point, we have put it up here here in this case here and here in this case here. So the points are in the domain or the values of the numbers and there is no derivative. So they are critical numbers. So we had these two anomalies near the 10 points of the domain in order to Add two more critical numbers but which are not local maximum nor local minimum. So that's the idea behind that. So I take this out a bit, so this is an example and solving for P. And we had this a little bit easier in for A. And that's it.