Section 3: Extrema and Models

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 6.3 Extrema and Models 583 Version: Fall 2007 6.3 Extrema and Models In the last section, we used end-behavior and zeros to sketch the graph of a given polynomial. We also mentioned that it takes a semester of calculus to learn an analytic technique used to calculate the “turning points” of the polynomial. That said, we’ll still pursue the coordinates of the “turning points” in this section, but we will use the graphing calculator to assist us in this quest; and then we will use this technique with some applications. Extrema Before we begin, we’d first like to differentiate between local extrema and absolute extrema . 2 This is best accomplished by means of an example. Consider, if you will, the graphs of three polynomial functions in Figure 1 . In the first figure, Figure 1 (a), the point A is the “absolute” lowest point on the graph. Therefore, the y -value of point A is an absolute minimum value of the function. In the second figure, Figure 1 (b), there is no “absolute” highest point on the graph (the graph goes up to positive infinity), nor is there an “absolute” lowest point on the graph (the graph goes down to negative infinity). Therefore, this function has neither an absolute minimum nor an absolute maximum . However, point B in Figure 1 (b) is the highest point in its immediate neighbor- hood. If you wander too far to the right, there are points on the graph higher than point B , but locally point B is the highest point. Therefore, the y -value of point B is called a local maximum value of the function. Similarly, point C in Figure 1 (b) is the lowest point in its immediate neighborhood. If you wander too far to the left, there are points on the graph lower than point C , but, in its neighborhood, point C is the lowest point. Therefore, the y -value of point C is called a local minimum of the function. x y A x y B C x y D E F (a) (b) (c) Figure 1. Differentiating between local and absolute extrema. Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 The term extrema , the plural of extremum , is a mathematical term that is used to refer to absolute 2 or local maxima or minima of a function. Note: Some mathematicians prefer the words global and relative to the words absolute and local. They are equivalent.
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584 Chapter 6 Polynomial Functions Version: Fall 2007 Finally, take a look at the graph in Figure 1 (c). Point F is the “absolute” lowest point on the graph, so the y -value of point F is an absolute minimum of the function. On the other hand, there is no highest point on the graph in Figure 1 (c), as each end of the graph escapes to positive infinity. Hence, the function has no absolute maximum. Locally, point D in Figure 1 (c) is the lowest point, so the y -value of point D is a local minimum of the function. Similarly, in its immediate neighborhood, point E is the highest point, so the y -value of point E is a local maximum.
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