Transcript Absolute Extrema

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
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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
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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
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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
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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
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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]
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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
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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
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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
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Assignment
Lesson 6.1
Page 372
Exercises 1 – 53 odd
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