90 CHAPTER 2 Functions and Their Graphs Determining Where a Function Is Increasing, Decreasing, or Constant from Its Graph Determine the values of x for which the function in Figure 28 is increasing. Where is it decreasing? Where is it constant? EXAMPLE 3 Solution When determining where a function is increasing, where it is decreasing, and where it is constant, report the largest interval(s) that satisfy the definition involving the independent variable x. The function whose graph is given in Figure 28 is decreasing on the interval 6, 4 [ ] − − because for any choice of x1 and x2 in the interval for which x x , 1 2 < we have f x f x . 1 2 ( ) ( ) > For example, 6− is included in the interval where f is decreasing because if x is any number for which x 6 4, − < ≤− then f f x 6 . ( ) ( ) − > Similarly, f is increasing on the interval 4, 0 [ ] − because for any choice of x1 and x2 in the interval for which x x , 1 2 < we have f x f x . 1 2 ( ) ( ) < Finally, the function f is constant on the interval 0, 3 [ ] and is decreasing on the interval 3, 6 . [ ] Figure 28 x y 6 5 22 24 (3, 4) (6, 1) (0, 4) (22, 0) (26, 0) (24, 22) y = f(x) Figure 27 x y f(x1) l x1 x2 f(x2) (a) For x1 , x2 in l, f(x1) , f(x2); f is increasing on I. x y f(x1) l x1 x2 f(x2) (b) For x1 , x2 in l, f(x1) . f(x2); f is decreasing on I. x y f(x1) x1 x2 f(x2) (c) For all x in I, the values of f are equal; f is constant on I. l height. The interval I on which a function is increasing, decreasing, or constant may be open, closed, or half-open/half-closed depending on whether the endpoints of the interval satisfy the required inequality or not. CAUTION Describe the behavior of a graph in terms of its x-values. Do not say the graph in Figure 28 is increasing from the point ( ) − − 4, 2 to the point ( ) 0, 4 . Rather, say it is increasing on the interval [ ] −4, 0 . j Now Work PROBLEMS 13, 15, 17, AND 25(c) 4 Use a Graph to Locate Local Maxima and Local Minima Suppose f is a function defined on an open interval I containing c. If the value of f at c is greater than or equal to the other values of f on I, then f has a local maximum at c.† See Figure 29(a). If the value of f at c is less than or equal to the other values of f on I, then f has a local minimum at c. See Figure 29(b). Figure 29 Local maximum and local minimum y y (a) f has a local maximum at c. (b) f has a local minimum at c. c x f (c) f (c) (c, f (c)) (c, f (c)) c x l l NOTE “Maxima” is the plural of “maximum”; “minima” is the plural of “minimum.” j †Some texts use the term relative instead of local.
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