00:01
We want to explain the difference between an absolute minimum and a local minimum.
00:08
Well, basically, we are talking here about where we look at the function to talk about an absolute minimum and a local minimum.
00:22
So, when we talk about an absolute minimum, talk about the image of the value where we have an absolute minimum is less than an absolute minimum, than or equal to all the values of the function in the domain of the function.
00:56
That is, the image of that point, or the point c, where we have a lot, 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 in the domain of the function.
01:15
So is the smallest of the images of the function looking at all the images, in the domain of the function.
01:25
When we talk about a local minimum, we are only looking at values of the function near some value c.
01:35
For example, we say f of c is less than or equal to f of x for all x near the value c.
01:47
That is, we normally take an interval, generally open interval, which contains the value z here, the value c, and then at that interval, we find that all the images are greater than or equal to the image of that value c, and so then we are talking about a local minimum.
02:14
But obviously, when we restrict, constraint, or change, or redefine the domain of the function, these definitions can overlap some way.
02:26
What is important is when we have defined the domain of the 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 is the absolute minimum.
02:46
There's no doubt about it.
02:47
But in the case of local minimum, we got to see only near the value where the function attains that local or minimum.
02:58
So it's better to put some examples.
03:04
And for example, we can see this function here.
03:09
Let's say, and let's say that the function has this domain.
03:17
I'm drawing here, that is, the domain is this close interval ab, let's say.
03:26
So we can see that the smallest of the images of this function happen here at the the end left end point.
03:38
That is, if we draw a horizontal line, we see that that image, f of a, is an absolute minimal, because it's the smallest of the images of all the points in the domain, in this case, in the interval ab.
04:05
But we can also see that this value here, the image of this point is 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.
04:40
There, there, these image, the image of point c is the smallest of the image, let's say, indecitable i, it's called it i.
04:52
And for that reason, at c we have a local minimum.
04:58
We cannot say, for example, that at this point we have a local minimum.
05:03
It's not true that if we look close to this point only, the image of that point will be the smallest of the minimum.
05:12
At the contrary, this is, at this point we have a local maximum.
05:19
For example, let's put another interval here.
05:23
Let's say we stay at this interval and we look at the function there.
05:32
As you can see, there is no local minimum in that open interval.
05:39
That's why we have a function that is increasing in that interval, and because the interval is open, there is no smallest nor largest value.
05:53
There can be also situations like this.
06:00
Let's put this graph here that resembles a little bit to looks like the absolute value function of x.
06:10
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.
06:24
It's the image of zero.
06:26
That is right here, f of zero is an absolute minimum, because if you look at all the domain, suppose the old domain is the whole real line, it is the smallest of the values.
06:40
It's an absolute minimum.
06:42
But it's also a local minimum.
06:43
If you look close to that point only, that is you stayed at a point.
06:49
Proportion as a graph that contains zero, of course, in that case is, again, the smallest of the values...