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

Experimentwith Excelspreadsheet

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 mayhave 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

• … for all x in the interval

Similarly f(c) is the absolute maximum if for all x in the interval

4

( ) ( )f x f c

( ) ( )f x f c

c

f(c)

Reminder – the absolute max or min is a y-value,

not an x-value

Reminder – the absolute max or min is a y-value,

not an x-value

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, minimums5

Strategy

To find absolute extrema for f on [a, b]

1. Find all critical numbers for f in open interval (a, b)

2. Evaluate f for the critical numbers in (a, b)

3. Evaluate f(a), f(b) from [a, b]

4. Largest value from step 2 or 3 is absolute max

Smallest value is absolute min

6

Try It Out

For the functions and intervals given, determine the absolute max and min

7

4 2( ) 32 7 on [-5, 6]f x x x

8 on [4, 6]

8

xy

x

2/32( ) 18 on [-3, 3]f x x

Graphical Optimization

Consider a graph that shows production output as a function of hours of labor used

8

hours of labor

Out

put We seek the hours of labor

to use to maximize output per hour of labor.

We seek the hours of labor to use to maximize output

per hour of labor.

Graphical Optimization

For any point on the curve• x-coordinate measures hours of labor• y-coordinate measures output• Thus

9hours of labor

Out

put

output ( )

hours of labor

y f x

x x

We seek to maximize this

value

We seek to maximize this

value

Note that this is also the slope of the line from the origin through a

given point

Note that this is also the slope of the line from the origin through a

given point

Graphical Optimization

It can be shown that what we seek is the solution to the equation

10hours of labor

Out

put

( )'( )

f xf x

x

Now we have the (x, y) where the line through the origin and tangent

to the curve is the steepest

Now we have the (x, y) where the line through the origin and tangent

to the curve is the steepest

Assignment

Lesson 6.1

Page 372

Exercises 1 – 53 odd

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