Explain the difference between an absolute minimum and a local minimum.
A function $f$ has an absolute minimum at $x=c$ if $f(c)$ is the smallest function value on the entire domain of $f,$ whereas
$f$ has a local minimum at $c$ if $f(c)$ is the smallest function value when $x$ is near $c$
we want to explain the difference between an absolute minimum and a local minimum. Well, basically we are talking here about where we look at the function to talk about an absolute minimum. And then at a minimum. So when we talk about an absolute minimum, talk about the image of the value where we have a lot to an absolute minimum is less than or equal to all the values of the function into the men of the function. That is the image of that point or the point. See where we have a look an absolute minimum is the smallest value of the function. That is is less than or equal to all the images. But for all points into the male of the function. So is the smallest of the images of the function. Looking at all the images in the domain of the function. When we talk about a block, who minimum? We are only looking at values as a function near some value. See for example, you say FFC is less than a record F of X for all eggs nears the value. See that is We normally take an interval generally open interval which contains the values E here, the policy. And then at that interval we find that all the images are greater than or equal to the image of that policy. And so then we are talking about local minimum. Yeah. But obviously when we constrain or change or redefine the benefit of the function. This definitions can overlap some way. But what is important is when we have defined the domain function or the domain is implicitly defined when we know the domain of the function, then the smallest of all the images of the function he is the absolute minimum. There's no doubt about it. But in the case of local minimum we got to see only near the value where we where the function attains that local minimum. So it's better to good some examples. And for example, we can see this function here, let's say. And let's say that the function has this domain. I'm drawing here, that is the main is this close interval A B let's say. So we can we can see that the smallest of the images of this function have been here at the end left in point that is if we draw a horizontal line, we see that that image F. Of a is uh an absolute minimum. Because it's the smallest of the images of all the points in the domain. In this case, in the interval maybe. But we can also see that this value here, the image of this point, it's a local minimum. And that's because if we look at that point here, let's call it C. And we stay, for example, near that value. We restrain our attention constrain our attention to this interval containing C. And then the graph of the function is only this portion here there. They're these Seamus, the image of points, the east, the smallest of the image. Let's say in this interval i it's called. I And for that reason let's see, we have a local mm minimum. We cannot say for example that at this point we have a local minimum. It's not true that if we look close to this point only the image of that point will be the smallest. At the contrary. This is at this point we have a local maximum. For example, let's put another interval here. Let's say we stay at this interval and we look at the function there as you can see there is no a local minimum in that open intervals. That's why we have a function that is increasing that interval. And because the interval is open, there is no smallest nor largest volume. There can be also situations like this, let's put this graph here that resembles a little bit too looks like the absolute value function of eggs. And that graph as you can see if we look close to the image of zero, which is zero, it doesn't matter how close you look. The smallest value is zero Is the image of zero that is here, F0 is an absolute minimum. Because if you look at all the domains supposed to all the Minister whole re line, it is the smallest of the values. It's an absolute minimum. But it's also a local minimum. If you look close to that point only that is you you stayed at a portion of the graphic that contains zero. Of course in that case is again the small social policy that is in this case is absolute minimum and a local minimum. And when we have several local minima function on a domain, then uh the smallest of all those local minimum is equal to the absolute minimum or the global minimum. But as you can see we can have several local minimum but only one absolute minimum. For example, let's put another example, something like Yeah. Mhm. Okay, if we have for example the function like this, let's say that and the domain of the function is it's pretty here and the man, let's say is this interval here A B. So we have this value here is local minimum. If you look only around this value and some clothes, so hoping to are close to that value containing that value, we can see that that's the monetary values. So we have a local minimum here. We have and also a local minimum here. And we can say that at B But we want to talk about local minimum when we can put an interval containing an open interval containing the point where we say there is a local minimum. So we don't talk about and points because we cannot put an opening tool to the right, for example of B. There's no function there. So local name in our and inside the domain and no more local minimum, desire to welcome in but from all the graph that is if you look at the whole function and we choose the smallest of the values, then we have this absolute minimum. So here we have an absolute minimum. So at this point but at this point we have is a local minimum and also an absolute minimum. But in here at this point here we have only local minimum because the absolute minimums this year cannot be at this point. But we can have situations where we have several points where we had the local or absolute minimum because one of the examples we know very well is signing codes and functions. So here we have this function that is signed in this case if you consider the complete functions that you see domain is the real numbers, then we can say that the absolute maximum value of the function is one but it's attained several points. That's my problem. That can The theorem says that there is at least one. So in this case you can say that the function attains an absolute maximum value of one in this case uh infinitely infinitely many points but we also attained the absolute minimum negative one. And at infinitely many point. But the these points where we have an absolute maximum absolute minimum, they are also local maximum. Local minimum. That is this local minimum. Because if you look close to the point is the smallest value is a local maximum. The maximum. So there are, we can combine there are several examples where we have the situation. So if we look for example, another example here you will take a line let's say mhm And if we take a line like this increasing and we don't include the points. That is the domain is the open in general A B. Then there is no local minimum, no local maximum, no absolute minimum and no absolute maximum. Sometimes it's hard to understand that. But the thing is that if you say there is an absolute maximum value then we can find another one that could be an absolute maximum there is. When this is not included, we have the possibility of putting always a point in between what you say is an absolute maximum. So there's there's no absolute maximum. It's got to be closed in order to be sure that the absolute maximum for absolute minimum exists and uh they continue to use the other essential property that guarantees the existence of the absolute maximum and extra medium. So when we have a continuous function and a close interval there is always uh an absolute maximum and an absolute minimum. So I keep here, we don't have local minimum nor local. Sorry, absolutely, man, Okay, mm, so in summary the absolute minimum function is a value except where the scene, the domain of the function for is the image of that value is less than or equal to all the other images of all the points in the main function and the local minimum refers to the smallest of the value of the function, but not in the whole domain, but near the valley. See near means are relatively small interval around by the sea for general and open intervals around containing C or around, see, for which the image of sees the smallest of the images of that interval, and and that's the main difference.