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Chapter 3

Chapter 3 . 3.1 Extrema on an Interval. Relative maximum : The maximum value relative to a neighborhood. Relative minimum : . Absolute maximum : The function’s greatest value . Absolute minimum : The function’s smallest value.

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Chapter 3

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  1. Chapter 3 3.1 Extrema on an Interval

  2. Relative maximum : The maximum value relative to a neighborhood. Relative minimum: Absolute maximum: The function’s greatest value . Absolute minimum : The function’s smallest value. Extremaare the minimum or maximum values of a function. (Extremum is a single value)

  3. Absolute Extrema • When will a function have an absolute maximum or absolute minimum? has a hole at x=0. Domain: [-1,2] [-1,2] on (-1,2)

  4. Extreme Value Theorem (another super important theorem) If is continuous on a closed interval ], then will have both a maximum and a minimum on the interval.

  5. Definition of a Critical Number (point) A critical point of a function is any point where: 1) or 2) does not exist

  6. Relative (local) extrema and critical points The following is an existence theorem: If has a relative extremum at a point then the point must be a critical point. Extremum at a point  Point is a critical point Reverse is not always true.

  7. Finding Extrema of on a Closed Interval (a different method applies for open intervals) Find the critical numbers (or dne) Evaluate at each critical number Evaluate at each endpoint The least value is the minimum and largest is the maximum Ex. Find the extrema of

  8. Ex. Find the extrema of

  9. Ex. Find the extrema of

  10. Ex. Find the extrema of

  11. Homework: 3.1 p. 167 #3-15(odd), 19, 23, 25, 33, 39, 43, 53, 55, 57, 63, 65, 67 Login to : cengage.com/login Access code: CM-9781285076225-0000015 Textbook access, practice quizzes

  12. Does have a minimum in the open interval ?

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