(a) Sketch the graph of a function on $ [-1, 2] $ that has an absolute maximum but no local maximum.
(b) Sketch the graph of a function on $ [-1, 2] $ that has a local maximum but no absolute maximum.
Graphs shown in the video
in this problem, we have two parts in part A. We are going to sketch the graph of function on negative 12 that has an absolute maximum but no local maximum in probably. We sketch the graph of function on the same interval negative one to that has a local maximum but no absolute maximum. So let's start with a We have to define or draw a function whose domain is nearly 12. We have these sketch here, We have then the domain is from negative 1-1, Sorry, from 91- two. Old points there has image and we can see this decreasing simply decreasing function. Um and those for these functions, the key thing is that it is always increasing them for that reason there are no uh local maximum. So we have this part fulfilled, there is no local maximum and we have an absolute maximum because that happens here at that point. Yeah, That is the absolute maximum is the image of 91 but there is no local maximum because this value here, which is the absolute maximum course at the left end point. So it cannot be considered an a local maximum. And inside of the internal negative 12, there is no local maximum at all. So this is an example, this could be also a line but you can draw a it's curved like this having always the decreasing behavior in the whole interval. Now they want to and with that we have the result we want but the part piece is a little more interesting because we have to do the inverse thing, That is the same domain. But now the function has a local maximum, no absolute minimum key thing to notice is in both parts there is no mention about continuity of the function or what happened with the minimum or local minimum. So we have sketched the solution for papi, papi and you can see that to have a local maximum inside the interval like this point here here, but no absolute maximum. There's got to be a behavior like this. This point here, that should be the highest value in The graph to hide his point. It is not included in the graph but the image of -1 is put over here. So we have created discontinuity that has the effect of the function not having an absolute maximum value in the interval. That is because we had when we don't include this point means that the function can have always a greater value to create about it all the time where we are very close to negative one from the right, but he never reaches that high is one. Right. And that forces us to create this jump is continue to like this. Mhm But we have a local maximum at this point here. And let's look at the graph has to have at least two other uh local minimum this case, but there is no mention about that in the statements of there's no problem. And because there was no mention about continually of the function, we could do that to create the effect of having an absolute maximum for the function. But if we include the uh continually, it will be another problem. Maybe we cannot have solution in that case. But uh the key thing in this party is too think about at this continually, that creates the effect. So we have then a function here with absolute maximum, but no local maximum. And in Burpee, okay, function with one local maximum. No absolute maximum. Okay, that a solution to the given problem?