12/19/2023 0 Comments Make a sign chart calculusKeep in mind that not all of these extrema will exist for every function. A function can have infinitely many local/relative minima, but it’ll only one (or no) global/absolute minimum. If a local minimum also happens to be the function’s lowest point anywhere in its domain, then it’s also the global/absolute minimum. A function can have infinitely many local/relative maxima, but it’ll only one (or no) global/absolute maximum.Ī local/relative minimum exists wherever the function changes direction from decreasing to increasing. If a local maximum also happens to be the function’s highest point anywhere in its domain, then it’s also the global/absolute maximum. We wouldn’t define an absolute maximum for the function, because it shoots up toward ?\infty? both to the left of ?A? and to the right of ?C?, so there’s no finite point we could name that would describe the highest value that the function ever reaches.Ī local/relative maximum exists wherever the function changes direction from increasing to decreasing. The function also has a local maximum at ?B?, because ?B? is the function’s highest point in the area near ?B?. But the function also has a local minimum at ?C?, because ?C? is the function’s lowest point in the area near ?C?. We already said that the function’s absolute/global minimum is at ?A?, and that’s because ?A? is the point where the function has the lowest value over its entire domain. And within the function’s maxima, we distinguish between local (relative) maxima and global (relative) maxima.Īs an example, let’s look again at the same graph as before. Within the function’s minima, we distinguish between local (relative) minima and global (absolute) minima. The function’s extrema are made up of its least points, which we call the minima, and its greatest points, which we call the maxima. Think about the extrema as being the function’s “extreme points.” In general, a function’s least and greatest values are its extrema. We’ll finish the section by translating these optimization steps into learning how to sketch the function’s graph. In this section, we’re going to talk all about the optimization process, starting with how to find a function’s least and greatest values. It let’s us calculate the point at which a function is maximized or minimized, and that has all kinds of real-world applications, which we’ll talk about in depth later in the course. If I know how to do the math to calculate this value, then I’ll know the exact temperature at which I should set the freezer, in order to minimize the chance that my food will spoil.Īnd that’s really valuable! Getting this right can save me time and money, and help make sure that my restaurant runs smoothly and is successful in the long term. If I’m the business owner of this restaurant, and I want to minimize the likelihood that my food will spoil, then I’m very interested in finding this global minimum. Let’s pretend for a moment that the function shown in the graph actually models the likelihood that food will spoil in a restaurant freezer at varying temperatures. There’s a reason why it’s important that we be able to find this global minimum.
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