 Let's have a quick review over the main ideas of section 3.3 inactive calculus on global optimization. So back in section 3.1, we introduced two tests for identifying local extreme values of a function, but we didn't talk about finding global extreme values. And that's what this section is all about. The first thing to realize is that sometimes functions may have local extreme values without having any global extreme values. For example, this function has a local minimum and a local maximum value, but there are no global extreme values of any sort because the function continues to increase to the right and decrease to the left, so there is no absolute highest or lowest point. Therefore, we're not always guaranteed that global extreme values will exist for any given function. However, if we make our function satisfy some basic conditions, then we will get global extreme values. And that's spelled out in the extreme value theorem. This theorem, or a mathematical fact, says that if f is a continuous function on a closed interval from a to b, then f attains both an absolute minimum and an absolute maximum value on the interval from a to b. So in other words, if we require f to be continuous, that is no breaks or jumps or asymptotes in its graph, and stay confined on a closed interval that includes its endpoints, it must reach an absolute high point and an absolute low point somewhere, possibly right at the endpoints themselves. The simple fact has many ramifications for real-world problems that are explored in this section a bit later, as well as in great depth in the next section, 3.4. For now, let's recall the workflow that's spelled out in your textbook for finding global extreme values. First of all, the function whose global extreme values we are trying to find must be continuous and must be restricted to a closed interval. If not, then the following steps won't always work. But if so, then first of all, find all the critical values of f that lie inside the interval. These are the points, remember, where f prime of x is zero, or f prime of x is undefined. Evaluate, secondly, the function f at all those points you found in step one, and at the two endpoints of the interval. Then look through the list you just created in step two. The smallest of the values from step two is the global minimum value of f on the closed interval from a to b, and the largest of those is the global maximum value on f on the interval from a to b. In the upcoming videos, we'll see examples of this workflow put to use.