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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. Intuitive Definition.

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Absolute extrema l.jpg

Absolute Extrema

Lesson 6.1


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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


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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 mayhave several relative extrema

    • It never has more than one absolute max or min


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Reminder – the absolute max or min is a y-value, not an x-value

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

c

f(c)


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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

    • Smallest value is absolute min


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Try It Out

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


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Graphical Optimization

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

We seek the hours of labor to use to maximize output per hour of labor.

Output

hours of labor


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

Graphical Optimization

  • For any point on the curve

    • x-coordinate measures hours of labor

    • y-coordinate measures output

    • Thus

We seek to maximize this value

Output

hours of labor


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Graphical Optimization through a given point

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

Now we have the (x, y) where the line through the origin and tangent to the curve is the steepest

Output

hours of labor


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Assignment through a given point

  • Lesson 6.1

  • Page 372

  • Exercises 1 – 53 odd


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