Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 38. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Analysis of Extrema 351 i f x Ixl on -to ii f x x on -1 2 Figure Local Extrema Suppose x c is an interior critical point of a continuous function f. How can we tell if at x c f has a local maximum local minimum or neither One approach is to look at the sign of f to determine whether f changes from increasing to decreasing across x c. This type of analysis is referred to as the first derivative test. If f is continuous x c is an interior critical point of f and f is differentiable on an open interval around c even if f is not differentiable specifically at c then if f changes sign from negative to positive at x c then f has a local minimum at c if f changes sign from positive to negative at x c then f has a local maximum at c if f does not change sign across x c then f does not have a local extremum at c. graph of f sign of f - c local min at c graph of f sign of f c -local max at c Figure Suppose x c is an interior critical point of f but f is not continuous. You might wonder if there is a first derivative test we can apply. The answer is no. Look carefully at the graphs presented in Figure . Figure Global Extrema Suppose we ve rounded up the usual suspects for extrema we ve identified the critical points of . Is there an easy way to identify the global maximum and minimum values First we have to figure out whether or not the function has a global maximum. If we know it does 352 CHAPTER 10 Optimization we can calculate the value of the function at each of the critical points. The largest value is the global maximum value. The corresponding x gives you the absolute maximum point. Sometimes you can exclude a few candidates. For instance a local minimum will never be a global maximum. List all critical points of f. Critical point x f x Compare the values of f at its critical points. Some functions don t have absolute maximum and minimum values. Think about the functions x 1 x and h x x2 each on its natural domain. The former has neither a .
