SECTION 2.3 Properties of Functions 91 5 Use a Graph to Locate the Absolute Maximum and the Absolute Minimum Look at the graph of the function f given in Figure 31. The domain of f is the closed interval [ ] a b , . Also, the largest value of f is ( ) f u and the smallest value of f is ( ) f v . These are called, respectively, the absolute maximum and the absolute minimum of f on [ ] a b , . Figure 30 x y –2 3 2 (1, 2) (–1, 1) y 5 f(x) DEFINITIONS Local Maximum/Minimum Let f be a function defined on some interval I and let c be a number in I. • A function f has a local maximum at c if there is an open interval in I containing c so that f c f x ( ) ( ) ≥ for all x in this open interval. The number f c( ) is called a local maximum value of f . • A function f has a local minimum at c if there is an open interval in I containing c so that f c f x ( ) ( ) ≤ for all x in this open interval. The number f c( ) is called a local minimum value of f . Finding Local Maxima and Local Minima from the Graph of a Function and Determining Where the Function Is Increasing, Decreasing, or Constant Figure 30 shows the graph of a function f. (a) At what numbers x, if any, does f have a local maximum? List the local maximum value(s). (b) At what numbers x, if any, does f have a local minimum? List the local minimum value(s). (c) Find the intervals on which f is increasing. Find the intervals on which f is decreasing. Solution EXAMPLE 4 The domain of f is the set of real numbers. (a) f has a local maximum at 1, since for all x close to 1, we have ( ) ( ) ≤ f x f 1 . The local maximum value is ( ) = f 1 2. (b) f has local minima at −1 and at 3. The local minimum values are ( ) − = f 1 1 and ( ) = f 3 0. (c) The function whose graph is given in Figure 30 is increasing on the intervals [ ] −1, 1 and [ )∞ 3, , or for − ≤ ≤ x 1 1 and ≥ x 3.The function is decreasing on the intervals ( ] −∞ −, 1 and [ ] 1, 3 , or for ≤− x 1 and ≤ ≤ x 1 3. If f has a local maximum at c, then the value of f at c is greater than or equal to the values of f near c. If f has a local minimum at c, then the value of f at c is less than or equal to the values of f near c. The word local is used to suggest that it is only near c, not necessarily over the entire domain, that the value ( ) f c has these properties. Now Work PROBLEMS 19 AND 21 DEFINITIONS Absolute Maximum and Absolute Minimum Let f be a function defined on some interval I. • If there is a number u in I for which f u f x ( ) ( ) ≥ for all x in I, then f has an absolute maximum at u , and the number f u( ) is the absolute maximum of f on I . • If there is a number v in I for which f v f x ( ) ( ) ≤ for all x in I, then f has an absolute minimum at v , and the number f v( ) is the absolute minimum of f on I . The absolute maximum and absolute minimum of a function f are sometimes called the absolute extrema or the extreme values of f on I. The absolute maximum or absolute minimum of a function f may not exist. Let’s look at some examples. NOTE The requirement that the number c be in an open interval means that a local maximum or a local minimum will never occur at an endpoint. j CAUTION The y -value is the local maximum value or local minimum value, and it occurs at some number x. For example, in Figure 30, we say f has a local maximum at 1 and the local maximum value is 2. j Figure 31 domain: [a, b] for all x in [a, b], f (x) # f (u) for all x in [a, b], f (x) $ f (v) absolute maximum: f (u) absolute minimum: f (v) (u, f(u)) (a, f(a)) y 5 f(x) (b, f(b)) (v, f(v)) a u v b x y
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