1. 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: Decrease: (-∞, -2.4) (-2.4, 1.6) (1.6, ∞)

2. Objectives: 1.Be able to define various vocabulary terms needed to be successful in this unit. 2.Be able to understand the definition of extrema of a function on an interval and “The Extreme Value Theorem”. 3.Be able to find the relative extrema and critical numbers of a function. 4.Be able to find extrema on a closed interval. Critical Vocabulary: Extrema, The Extreme Value Theorem, Critical Numbers

3. 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 IntervalClosed Interval

4. 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) = x 2 + 3, Interval: [-2, 2], Let c = 0 f(c): f(0) = 3 f(-2) = 7f(-1) = 4f(1) = 4f(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

5. 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) = x 2 + 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)

6. 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 Min “Valley” Relative Max “Hill” Relative Min “Valley” Point where a graph changes its behavior (increasing/decreasing) help in determining the maximum and minimum values of a graph. III. Relative Extrema and Critical Numbers

7. 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. III. Relative Extrema and Critical Numbers 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.

8. Example 2: Find the value of the derivative at each of the relative extrema shown in the graph When the derivative is zero, we call the x-value associated with it a CRITICAL NUMBER.

9. III. Relative Extrema and Critical Numbers Example 3: Find any critical numbers algebraically: f(x) = x 2 (x 2 - 4) 1 st : Find the derivative 2 nd : Set f’(x) = 0 3 rd : Check for any places where the derivative is undefined

10. IV. Finding Extrema on a Closed Interval 1. Find the critical numbers of f in (a, b) To find the extrema of a continuous function f on a closed interval [a, b], use the following steps: 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.

11. IV. Finding Extrema on a Closed Interval Example 4: Locate the absolute extrema of the function on the closed interval 1 st : Find the critical numbers 2 nd : Evaluate at the endpoints Left End point:Right End point: Minimum Maximum

12. IV. Finding Extrema on a Closed Interval Example 5: Locate the absolute extrema of the function on the closed interval 1 st : Find the critical numbers Critical number: x = 0 2 nd : Evaluate at the endpoints Left End point:Right End point: MinimumMaximum

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14. 1. Find all the relative extrema of the function f(x) = x 4 – 8x 2 2. Find the absolute extrema of the function: 3. Find the absolute extrema of the function: Practice Assessment

15. 1. Find all the relative extrema of the function f(x) = x 4 – 8x 2 Practice Assessment Relative Minimum Relative Maximum Relative Minimum

16. 2. Find the absolute extrema of the function: Practice Assessment Maximum Minimum

17. 3. Find the absolute extrema of the function: Practice Assessment Maximum Minimum