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Determine where a function is increasing or decreasing When determining if a graph is increasing or decreasing we always start from left and use only the x values. Included and excluded do not apply, we always use ( ). Increase: (-∞, -2.4) (1.6, ∞) Decrease: (-2.4, 1.6) 1. 2. 3. 4. Objectives: Be able to define various vocabulary terms needed to be successful in this unit. Be able to understand the definition of extrema of a function on an interval and “The Extreme Value Theorem”. Be able to find the relative extrema and critical numbers of a function. Be able to find extrema on a closed interval. Critical Vocabulary: Extrema, The Extreme Value Theorem, Critical Numbers I. Vocabulary Extrema (Plural of Extreme): This means we are talking about maxima (plural of maximum) and minima (plural of minimum) of a function. Interval: Means we are talking about a part of a function, denoted by interval notation (a, b) or [a, b] Open Interval Closed Interval II. Extrema of a Function Let f be defined on an interval I containing c 1. f(c) is the MINIMUM of f on I if f(c) ≤ f(x) for all x in I 2. f(c) is the MAXIMUM of f on I if f(c) ≥ f(x) for all x in I The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval. Example 1: Function: f(x) = x2 + 3, Interval: [-2, 2], Let c = 0 f(c): f(0) = 3 f(-2) = 7 f(-1) = 4 f(1) = 4 f(2) = 7 f(c) is less than or equal to all values of f(x) on the interval making f(c) the MINIMUM of f II. Extrema of a Function If f is continuous on a closed interval [a, b], then f has both a minimum and maximum on the interval. To illustrate this, we will look at the graph of f(x) = x2 + 1 on the following intervals: [-1, 2] Min: (0, 1) Max: (2, 5) (-1, 2) Min: (0, 1) No Max [-1, 2] No Min Max: (2, 5) III. Relative Extrema and Critical Numbers Point where a graph changes its behavior (increasing/decreasing) help in determining the maximum and minimum values of a graph. Relative Minimum: The smallest point of the graph in a given area. Relative Maximum: The largest point of the graph in a given area. Relative Max “Hill” Relative Max “Hill” Relative Min “Valley” Relative Min “Valley” III. Relative Extrema and Critical Numbers 1. If there is an open interval containing c on which f(c) is a maximum, then f (c) is called a relative maximum of f. 2. If there is an open interval containing c on which f(c) is a minimum, then f (c) is called a relative minimum of f. The plural of relative maximum is relative maxima and the plural of relative minimum is relative minima. III. Relative Extrema and Critical Numbers Example 2: Find the value of the derivative at each of the relative extrema shown in the graph f ( x) 9 x 2 27 x 3 f ( x) 9 x 1 27 x 3 f ' ( x) 9 x 2 81x 4 f ' ( x) 9 81 4 2 x x 9 x 2 81 f ' ( x) x4 9 x2 9 f ' ( x) x4 9 (3) 2 9 f ' (3) (3) 4 f ' (3) 0 9 x2 3 f ( x) x3 When the derivative is zero, we call the x-value associated with it a CRITICAL NUMBER. III. Relative Extrema and Critical Numbers Example 3: Find any critical numbers algebraically: f(x) = x2(x2 - 4) 1st: Find the derivative f ( x) x 2 x 2 4 f ( x) x 4 4 x 2 f ' ( x) 4 x 3 8 x 2nd: Set f’(x) = 0 0 4 x3 8x 0 4x x2 2 4x 0 x0 x2 2 0 x2 2 x 2 3rd: Check for any places where the derivative is undefined IV. Finding Extrema on a Closed Interval To find the extrema of a continuous function f on a closed interval [a, b], use the following steps: 1. Find the critical numbers of f in (a, b) 2. Evaluate f at each critical number in (a, b) 3. Evaluate f at each end point in [a, b] 4. The least of these values is the minimum and the greatest is the maximum. IV. Finding Extrema on a Closed Interval Example 4: Locate the absolute extrema of the function on the x2 closed interval f ( x) x2 3 1st: Find the critical numbers (0,0) f ( x) x 2 x 2 3 g ( x) x 2 , [1,1] Minimum 1 g ' ( x) 2 x h( x) x 2 3 1 h( x) x 2 3 2 x 2 f ' ( x) 2 xx 3 2 x x 3 1 2 3 f ' ( x) 2 xx 2 3 x 2 3 x 2 2 f ' ( x) 2nd 1 2 6x 6x 0 x 3 x 0 : Evaluate at the endpoints 1 Left End point: 1, 4 2 2 1 4 Right End point: 1, Maximum IV. Finding Extrema on a Closed Interval Example 5: Locate the absolute extrema of the function on the closed interval f ( x) x1/ 3 , [1,1] 1st: Find the critical numbers (0,0) f ( x) f ( x) 1 2 3 x 3 1 33 x 2 Critical number: x = 0 2nd : Evaluate at the endpoints Left End point: 1,1 Minimum Right End point: 1,1 Maximum Page 319-321 #7-27 odd, 33, 43, 45, 47 Practice Assessment 1. Find all the relative extrema of the function f(x) = x4 – 8x2 2 3 3 2 x x 9 x 2, [3,5] 3 2 x f ( x) 2 , [3,3] x 1 2. Find the absolute extrema of the function: f ( x) 3. Find the absolute extrema of the function: Practice Assessment 1. Find all the relative extrema of the function f(x) = x4 – 8x2 f ' ( x) 4 x 3 16 x x 0 : (0,0) x 2 : (2,16) Relative Maximum Relative Minimum x 2 : 2,16 Relative Minimum Practice Assessment 2 3 2. Find the absolute extrema of the function: f ( x) x 3 x 2 9 x 2, [3,5] 3 f ' ( x) 2 x 2 3 x 9 x 3 : 3, 41 2 Minimum x 3 2 : ( 3 2 , 79 8) Maximum x 3 : 3, 5 2 x 5 : 5,17 6 2 Practice Assessment 3. Find the absolute extrema of the function: f ' ( x) x2 1 x 2 x 1 : 1,1 2 x 1: (1,1 2) x 3 : 3, 3 10 x 3 : 3, 3 10 1 2 Maximum Minimum f ( x) x , [3,3] 2 x 1