Absolute Extrema
Lesson 6.1
Fencing the Maximum
You have 500 feet of fencing to build a
rectangular pen.
What are the dimensions which give you
the most area of the pen
Experiment
with Excel
spreadsheet
2
Intuitive Definition
Absolute max or min is the largest/smallest
possible value of the function
Absolute extrema often coincide with
relative extrema
A function may
have several
relative extrema
• It never has more than one absolute max or min
3
Formal Definition
Given f(x) defined on interval
• The number c belongs to the interval
Then f(c) is the absolute minimum of f on
the interval if
f ( x )  f (c )
• …
Reminder – the absolute
formax
allorxmininisthe
interval
a y-value,
not an x-value
c
f(c)
Similarly f(c) is the absolute maximum if
f ( x)  f (c) for all x in the interval
4
Functions on Closed Interval
Extreme Value Theorem
• A function f on continuous close interval [a, b]
will have both an absolute max and min on the
interval
Find all absolute maximums, minimums
5
Strategy
To find absolute extrema for f on [a, b]
Find all critical numbers for f in open
interval (a, b)
Evaluate f for the critical numbers in (a, b)
Evaluate f(a), f(b) from [a, b]
Largest value from step 2 or 3 is absolute
max
1.
2.
3.
4.

Smallest value is absolute min
6
Try It Out
For the functions and intervals given,
determine the absolute max and min
f ( x)  x 4  32 x 2  7 on [-5, 6]
8 x
y
8 x
f ( x)   x  18
2
2/ 3
on [4, 6]
on [-3, 3]
7
Graphical Optimization
Consider a graph that shows production
output as a function of hours of labor used
Output
We seek the hours of labor
to use to maximize output
per hour of labor.
hours of labor
8
Graphical Optimization
For any point on the curve
• x-coordinate measures hours of labor
• y-coordinate measures output
y
output
f ( x)
• Thus


x hours of labor
x
We seek to
maximize this
value
Output
Note that this is
also the slope of
the line from the
origin through a
given point
hours of labor
9
Graphical Optimization
It can be shown that what we seek is the
solution to the equation
f ( x)
f '( x) 
x
Output
Now we have the (x, y)
where the line through
the origin and tangent
to the curve is the
steepest
hours of labor
10
Assignment
Lesson 6.1
Page 372
Exercises 1 – 53 odd
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