Mooculus 8: Optimization
11 videos • 6,944 views • by Jim Fowler Last week, we started studying applications of the derivative: we continue applications this week, focusing on optimization problems. There's a wide variety of optimization problems that calculus will help us solve. These "word problems" can be quite complicated: not only can the calculus be tricky, but just translating the word problem into mathematics can be very hard. But if you're about to set out on a quest, it helps to know whether your quest will be successful! The Extreme Value Theorem states the following: if a function f is continuous on the closed interval [a,b], then there are points c and d in [a,b] so that for all x in [a,b], it is the case that f(c) ≤ f(x) and f(x) ≤ f(d). This looks intimidating! The upshot is that continuous functions on closed intervals attain their maximum and minimum value, so, yes, in many cases our search for the "optimum" value is not a fool's errand! There's a process for finding the maximum and minimum values of a function f on a given domain. You should differentiate f, find critical points, check critical points and endpoints, and check the behavior near the left and right hand endpoints, if you're working over, say, an open interval. Key points are to remember to check the endpoints and to remember that critical points include points of nondifferentiability.