Transcript 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.

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 …
f (c)  f ( x) x  D
Absolute Max/Min
D  (a, b
f(x)
a
b
• 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
Absolute Max/Min
• There will exist an absolute max/min for
 a continuous function
[a,b]
• Sometimes it is at the end points
• Some times it is on a peak or valley
 on a closed interval
•
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
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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) = 10 + 6x – x2
on [-4, 4]
• g(t) = 3t5 – 20 t3 on [-1, 2]
• k(u) = cos u – sin u on [0, 2]
•
f ( x)  x ( x  5)
1
3
on [0, 4]
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)
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solution
 we wish to maximize this function
Assignment
• Lesson 4.1
• Page 209
• Exercises 1 – 57 EOO
Also #71