3.1 Extrema on an Interval

1. 3.1 Extrema on an Interval Greg Kelly, Hanford High School Richland, Washington

2. Objectives • Understand the definition of extrema of a function on an interval. • Understand the definition of relative extrema of a function on an open interval. • Find extrema on a closed interval.

3. The mileage of a certain car can be approximated by: At what speed should you drive the car to obtain the best gas mileage? Of course, this problem isn’t entirely realistic, since it is unlikely that you would have an equation like this for your car.

4. We could solve the problem graphically:

5. We could solve the problem graphically: On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

6. The car will get approximately 32 miles per gallon when driven at 38.6 miles per hour. We could solve the problem graphically: On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

7. Notice that at the top of the curve, the horizontal tangent has a slope of zero. Traditionally, this fact has been used both as an aid to graphing by hand and as a method to find maximum (and minimum) values of functions.

8. Absolute extreme values are either maximum or minimum points on a curve. They are also sometimes called absolute extrema. (Extrema is the plural of the Latin extremum.) Definition of Extrema: Let f be defined on an interval I containing c. 1. f(c) is the min of f on I if f(c)≤ f(x) for all x in I 2. f(c) is the max of f on I if f(c) ≥ f(x) for all x in I.

9. Extreme values can be in the interior or the end points of a function. No Absolute Maximum Absolute Minimum

10. Absolute Maximum Absolute Minimum

11. Absolute Maximum No Minimum

12. No Maximum No Minimum

13. Extreme Value Theorem: If f is continuous over a closed interval, then f has a maximum and minimum value over that interval. Maximum & minimum at interior points Maximum & minimum at endpoints Maximum at interior point, minimum at endpoint

14. Relative Extreme Values: A relative maximum is the maximum value within some open interval. A relative minimum is the minimum value within some open interval.

15. Relative extremes are also called local extremes. Absolute maximum (also relative maximum) Relative maximum Relative minimum Relative minimum Absolute minimum (also relative minimum)

16. Notice that relative extremes in the interior of the function occur where is zero or is undefined. Absolute maximum (also local maximum) Local maximum Local minimum

17. Find the value of the derivative at the given extrema.

18. Find the value of the derivative at the extrema.

19. Relative Extreme Values: If a function f has a relative maximum value or a relative minimum value at an interior point c of its domain, and if exists at c, then

20. Critical Point: A point in the domain of a function f at which or does not exist is a critical point of f . Note: Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

21. 1 4 Find the derivative of the function, and determine where the derivative is zero or undefined in (a,b). These are the critical points. The least of these values is the minimum. The greatest of these is the maximum. 2 Find the value of the function at each critical point. 3 Evaluate f at each endpoint of [a,b]. Finding Maximums and Minimums on a Closed Interval [a,b]:

22. Critical points are not always extremes! (not an extreme)

23. (not an extreme)

24. EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval . Critical #s: x=0,1 Minimum We also must check the endpoints. Maximum

25. Minimum Maximum EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval . Critical #s: x=0,1

26. EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval .

27. Maximum Minimum EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval .

28. Homework 3.1 (page 165) #1-25 odd #26, 27, 29, #33-39 odd, #51