In this problem, we're asked to identify the extreme values of the function f(x) on the specified domain. Now, this is actually the same exact function in all of these graphs. The difference is the domain over which this function is defined. So let's go ahead and start all the way over on the left here with our first graph. Now looking at this function, it is defined over the domain of all real numbers from negative infinity to positive infinity.
So, what extreme values, what global maxima, and minima or local maxima, and minima does this function have? Well, looking at this function, I can see that the absolute lowest point on my graph is down here at x equals 0. Now, because this is the lowest point on my function, I can say that this is a global minimum. Now also, because this point is not an endpoint and I can kind of zoom in here, I can see that this is the lowest point compared to all nearby points. So, this is not only a global minimum, but it is also a local minimum.
So, we have a global and local minimum of 0 at x equals 0. This is our global and local minimum point. Now, when thinking about maximum values, looking at this function, since it's defined from negative infinity to positive infinity, I can see that my graph continues going off to infinity on both sides of this function, meaning that there's not one single maximum value that this function reaches. So, there are no maximum values for this function. There's only a minimum.
Let's move on to our next graph here. Here, we have the same function but now defined over the domain from 0 to 3, with both of those endpoints, 0 and 3, included in that domain. So, what are our maximum and minimum values here? Well, I can see that my very lowest point on my function is here at x equals 0. So, this point represents my global minimum.
Now, you may be thinking here, is this also a local minimum? But remember, because of the convention that we use in this course, endpoints cannot be local extrema. So, this point is only a global minimum. So, we have our global minimum of 0 at x equals 0. Now remember, this is something that may vary between your textbook or professor.
So, again, be sure to verify what convention you should use when it comes to considering endpoints as local extrema. Now, what about the other side of our function here? I can see that this very highest point happens here at x equals 3. So, this is again an endpoint. It can't be a local maximum, but it can be my global maximum, and it is because it's that very highest point. So, this function has a global maximum of 9, and it reaches that point at x equals 3.
So, this function has both a global minimum and a global maximum, but no local extrema because those endpoints were my global min and max. Now, let's move on to our last function here. Here, we see that our domain is defined from 0 to 3, but this time, my endpoints are not included in that domain. So, what are the extreme values of this function? Well, when we think about reaching the very lowest point, that would be our global minimum, we would think about this point here at 0.
But, this point is not included in our domain, so we can't say that that's where our global minimum is. Now, you might be thinking, well, couldn't our global minimum be at 0.01 or 0.001? That's exactly the problem that we're encountering here. We can get closer and closer to 0, but we can never quite reach it.
And there's always going to be a value on our function that is smaller than whatever we're thinking of. Even if I said 0.0001, you could still get smaller. So, there are no local or global extrema on this function because we come to the same conclusion when looking at this other point. We could say, okay, we could get to 2.99 or 2.999 to get to that maximum value.
But, remember that you can always get a higher value because we can't ever reach that endpoint because it's not included in our domain. So, for this last function here, there are no extreme values that we can identify. So, looking at these functions, we can see that even though they're all the same exact function, because they're defined over different domains, our extreme values look completely different. Now, be sure to keep this in mind as we continue practicing. I'll see you in the next one.
