Extrema on an Interval Lesson 4.1

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

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

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 f(x) a b

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

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

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 View animation of these concepts

Procedure 1. Determine f ' (x), set equal to zero 2. Solve for x (may be multiple values) 3. To find the point on the original function, substitute results back into f(x) 4. Note whether it is a max or a min by observing the graph or a table of values

Examples: f(x) = x – x 2 on [-4, 4] g(t) = 3t 5 – 20 t 3 on [-1, 2] k(u) = cos u – sin u on [0, 2  ] on [0, 4]

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

Assignment Lesson 4.1 Page 209 Exercises 1 – 57 EOO Also #71