Extrema on an Interval

1. Extrema on anInterval Lesson 4.1

2. Design Consultant Problem • A milk company wants to cut down on expenses • They decide that their milk carton design uses too much paper • For a given volume how can we minimize the amount of paper used? This lesson looks at finding maximum and minimum values of functions

3. Absolute Max/Min • Definition:f(x) is the absolute max (or min) on a set of numbers, D … if and only if …

4. f(x) b a Absolute Max/Min • Maximum is at b • There exist a value b such that f(b)  f(x) for all x in the interval • There is no minimum • No value, c exists so that f(c)  f(x) for all x • it is an open interval on the left

5. Absolute Max/Min • There will exist an absolute max/min for • a continuous function • on a closed interval [a,b] • Sometimes it is at the end points • Some times it is on a peak or valley •

6. Relative Max/Min • It is possible to have a relative max or min on an open interval • If so, it will be at a peak or valley

7. Relative Max/Min • Will be found at a place on the graph where: • f '(c) = 0 • or where f ‘(c) does not exist View animation of these concepts

8. Procedure • Determine f ' (x), set equal to zero • Solve for x (may be multiple values) • To find the point on the original function, substitute results back into f(x) • Note whether it is a max or a min by observing the graph or a table of values

9. Examples: • f(x) = 10 + 6x – x2 on [-4, 4] • g(t) = 3t5 – 20 t3 on [-1, 2] • k(u) = cos u – sin u on [0, 2] • on [0, 4]

10. Example: • Find two nonnegative numbers whose sum is 8.763 and the product of whose squares is as large as possible • The numbers are • x and (8.763 – x) • their product is x(8.763 - x) • we wish to maximize this function View Spreadsheet solution

11. Assignment • Lesson 4.1 • Page 209 • Exercises 1 – 57 EOOAlso #71