 Let's have a quick review over the main ideas of section 3.1 in active calculus on using derivatives to identify extreme values of a function. First of all, by extreme values of a function, we mean the highest and lowest points that the function attains. There are two ways to think about these extreme values, locally and globally. Given a function f and a point c in the domain of f, we say that f of c is a local or relative maximum if f of c is greater than or equal to f of x for all x near c. And likewise, f of c is a local or relative minimum if f of c is less than or equal to f of x for all x near c. So f of c is a local or relative extreme value if it is the highest or lowest point in an area restricted to being close to c. There might be some point where f is higher or lower as we get far away from that point c. But near c, this is the highest point if f of c is a local maximum or lowest point if f of c is a local minimum. On the other hand, if f of c is bigger than or equal to f of x for all x, that is if f of c is the highest point on the graph at any point, not just near c, then we say that f of c is a global maximum. Likewise, if f of c is less than or equal to f of x for all x, that is f of c is the lowest point on the graph anywhere, then f of c is a global minimum. A key task that is central to most of the applications for this chapter is the ability to locate local and global extreme values of a function with a high degree of precision. To do this, we notice that whenever a function has a local extreme value, the derivative of the function is either zero or undefined there. So we'll call any point where f prime of c is zero or undefined a critical value of f. And the importance of critical values lies in the first derivative test, which says that if p is a critical value of a continuous function that is differentiable near p, but not necessarily right at p. Then f has a relative maximum at p if and only if f prime changes sign from positive to negative at p. And f has a relative minimum at p if and only if f prime changes sign from negative to positive at p. This is a test in the sense that we can use it to classify a critical number as a local extreme value or decide that it is not a local extreme value. Importantly, we'll note that not all critical numbers do lead to local extreme values. Another test for classifying local extreme values is the second derivative test which says that if p is a critical value of a continuous function f such that f prime of p is zero and f double prime of p is not equal to zero. Then f has a relative maximum at p if and only if f double prime of p is less than zero. And f has a relative minimum at p if and only if f double prime of p is greater than zero. In other words, if p is a critical number for f, but not a critical number that makes the derivative undefined, just that makes the derivative equal to zero. And if the second derivative of f at p is not zero, then f of p is a local maximum if f is concave down at p. And a local minimum if f is concave up at p. If the second derivative of f at p is equal to zero, then this test yields no information. So the workflow for finding and identifying local extreme values of a function f goes like this. First, we need to find the critical values of f by taking the derivative of f and finding where that derivative is zero and finding where it's undefined. Then we test each critical value by using either the first or second derivative tests. Both tests accomplish the same task. But if the second derivative of f is easier to compute, then often this is the simpler test to use. On the other hand, if f double prime is hard to compute, then the first derivative test is usually better. In actual practice, step two is accomplished by using a construction that we'll call a sign chart for either f prime or f double prime. And we're about to see some examples of this in the upcoming videos.