SECTION 2.3 Properties of Functions 93 For example, the graph of the function f given in Figure 32(a) on the previous page is continuous on the closed interval [ ] 0, 5 . The Extreme Value Theorem guarantees that f has extreme values on [ ] 0, 5 . To find them, we list • The values of f at the local extrema: ( ) ( ) = = f f 3 6, 4 4 • The values of f at the endpoints: ( ) ( ) = = f f 0 1, 5 5 The largest of these, 6, is the absolute maximum; the smallest of these, 1, is the absolute minimum. Notice that absolute extrema may occur at the endpoints of a function defined on a closed interval. However, local extrema cannot occur at an endpoint because an open interval cannot be constructed around the endpoint. So, in Figure 32(b) on the previous page, for example, ( ) = f 1 2 is not a local maximum. Now Work PROBLEM 49 6 Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing To locate the exact value at which a function f has a local maximum or a local minimum usually requires calculus. However, a graphing utility may be used to approximate these values using the MAXIMUM and MINIMUM features. Now Work PROBLEM 57 7 Find the Average Rate of Change of a Function In Section 1.5, we said that the slope of a line can be interpreted as the average rate of change. To find the average rate of change of a function between any two points on its graph, calculate the slope of the line containing the two points. Using a Graphing Utility to Approximate Local Maxima and Minima and to Determine Where a Function Is Increasing or Decreasing (a) Use a graphing utility to graph ( ) = − + f x x x 6 12 5 3 for − ≤ ≤ x 2 2. Approximate where f has a local maximum and where f has a local minimum. (b) Determine where f is increasing and where it is decreasing. EXAMPLE 6 Solution (a) Graphing utilities have a feature that finds the maximum or minimum point of a graph within a given interval. Graph the function f for − ≤ ≤ x 2 2. Using MAXIMUM on a TI-84 Plus CE, we find that the local maximum value is 11.53 and that it occurs at = − x 0.82, rounded to two decimal places. See Figure 33(a). Using MINIMUM, we find that the local minimum value is −1.53 and that it occurs at = x 0.82, rounded to two decimal places. See Figure 33(b). Notice in Figure 33(c) that the local maximum and local minimum are identified on the graph drawn in Desmos. (c) Local extrema (b) Local minimum −10 −2 2 30 −10 −2 2 30 (a) Local maximum Figure 33 (b) Looking at Figure 33, we see that f is increasing on the intervals [ ] − − 2, 0.82 and [ ] 0.82, 2 , or for − ≤ ≤− x 2 0.82 and ≤ ≤ x 0.82 2. And f is decreasing on the interval [ ] −0.82, 0.82 , or for − ≤ ≤ x 0.82 0.82.
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