1. Absolute Extrema Lesson 6.1

2. 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

3. 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

4. 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 c f(c) Reminder – the absolute max or min is a y-value, not an x-value

5. 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

6. 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

7. Try It Out For the functions and intervals given, determine the absolute max and min 7

8. Graphical Optimization Consider a graph that shows production output as a function of hours of labor used 8 hours of labor Output We seek the hours of labor to use to maximize output per hour of labor.

9. Graphical Optimization For any point on the curve x-coordinate measures hours of labor y-coordinate measures output Thus 9 hours of labor Output We seek to maximize this value Note that this is also the slope of the line from the origin through a given point

10. Graphical Optimization It can be shown that what we seek is the solution to the equation 10 hours of labor Output Now we have the (x, y) where the line through the origin and tangent to the curve is the steepest

11. Assignment Lesson 6.1 Page 372 Exercises 1 – 53 odd 11