For each of the numbers $ a $, $ b $, $ c $, $ d $, $ r $, and $ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.
Absolute maximum at $s,$ absolute minimum at $r,$ local maximum at $c,$ local minima at $b$ and $r,$ neither a maximum nor a minimum at $a$ and $d$
in this problem, we are going to state whether the following function has an absolute maximum or minimum. The local maximum or minimum or neither a maximum nor a minimum at the numbers A B c D E r s. And here is a sketch of the graph given him the textbook correspondent to these exercise. And the first thing we got to do, this type of exercise is resolved for two global extremes of the function. That is the absolute maximum and the absolute minimum. After that we go for the local extreme. So, um talking about the the absolute maximum. In that case we refer to the largest image of all the images in the domain. Function in this case is the main of the function is assumed to be what The graph suggests. That is from a two s. This interval from a two s. This is the main of the function in that domain. We gotta look at the largest of the images and in that case that happens at s because the high has find in the graph. Is this one here his point here? Now we look at the absolute minimum. Yeah. Mhm. In that response to the smallest of the images or correspondingly the lowest point in the graph and that corresponds in this case too, the much are so I see absolute me at remember we are not talking about the value here, but because we don't have callus on the Y axis christmas to all these numbers A B C D R N S. But we are talking where decor, that's why we're talking about, that's what we say at the value. So the absolute minimum, of course at are now we have already these absolute stream, the global streams of functions. Now we go for the local and let's talk about first to local maximum. So um we can say that at sea, we have a local maximum here. Mm And that's because uh if you look close to see that as we stay, for example, at this open interval around see and we restrain our attention on that portion of the graph. They hide his value or the largest image, that small interval around see is the image at sea. So we have a local maximum at sea. It is not the global tell me it is not the global maximum because we have already talked about the global, absolute maximum discourse, that? S because the image of s is larger than the image of C. And this is the difference between local and absolute. That is for the absolute streams, you got to look at the whole domain. So in this case, let's see, we have a local maximum inside the into a S. We don't have any other local maximum but we have local minimum, for example, we have local minimum uh are which in fact is the absolute minima but it's also a local minimum in the sense that if you look close to that uh point that is near art in an interval containing art and we look at that they're the graph only there, the smallest value is the image of art. So we have a local minimum and our we have also a local minimum. Uh B Okay. These father hero. And that is because he's we look close to be only the smallest value of that portion of the graph is the message be. So we have these uh we can maybe use some colors too error. That is we are going to use these green for the local minimum. Better. So is green here for the local minimum, which are R and B. So that we have a local minimum. And are we have this little and the local maximum, let's say, let's see, let's say it was cross like this And that's the only one and then A. D. Which is the other point that was given. And we got to talk about A also. But at the if you look around the there is in a close interval around the it doesn't matter how small or big is this interval? The value of the image. That is not. It's neither the largest image nor the smallest image. That's because we always have images that are less than the image of the and images that are larger than the image at the that's because basically the function is decreasing an interval around the Yeah, I'm sufficiently close to the so A D. We don't have neither. We have neither uh maximum. Lower minimum at the but at a what happens at a. So it's important to talk about that at a we have the same situation as the we cannot say we have a local maximum at a because we don't have craft to the left of me so we cannot look around a both sides. So we don't have neither maximum or minimum pay. So here at the end and also had a okay we can say that the absolute maximum s is not a local maximum and that's the same reason for why at a we don't have neither maximum or minimum but because we have an absolute maximum our at S we still with that better. But the only point remained to classify or top Caldas de which is neither a maximum very minimum there at a where is the same situation as the. Okay so these are all the points we needed to classify. So um this is the answer to this problem. Yeah.