Closed and bounded regions
The mathematical process of optimization involves finding the maximum or minimum value of a function over a region. If we want to find the maximum of a function of x on an interval and the function is differentiable on this interval, we will have two cases to consider. The maximum can a occur at a critical point or it can occur at the end point. Remember, a critical point means that the second derivative has a zero at that point. That’s one thing that can happen. And the other thing that can happen is that the maximum occurs at a boundary point. To illustrate these two scenarios, we can use graphs. For example 1, our function looks like this, and on the interval where it is positive, its maximum value is 0. This is a critical point because g'(x0)=0, where g'(x) is the first derivative of f(x). In example 2, our function looks like this and it has one critical point; however, it is not on the boundary of the interval (it’s in the interior). And in more complex situations, the pattern is the same. It’s just more difficult to identify critical points in a more complex system. [Graph][Graph] We should say something about this region. A bounded region is one that includes all points within it and only those points, such as a disk or a solid square. An unbounded region extends beyond its boundary, such as the whole right upper quadrant of the plane. When the region is bounded, it is an interval in one dimension, and the entire positive number line or all positive numbers in two dimensions. The region bounded by x and y is the one where both x and y are positive. In one dimension, a bounded region is an interval. In an unbounded region, it would be like the whole line or all of the positive axis. So these are the ones that are like intervals.
