Extreme Values of Functions

1. Extreme Values of Functions Chapter 5.1

2. Absolute (Global) Extreme Values • Up to now we have used the derivative in applications to find rates of change • However, we are not limited to the rate-of-change interpretation of the derivative • In this section you will learn how we can use derivatives to find extreme values of functions (that is maximum or minimum values)

3. Definition of Extreme Values on an Interval DEFINITION: Let be defined on an interval containing . • is the minimum of on if for all in • is the maximum of on if for all in The minimum and maximum of a function on an interval are the extreme values or extrema (plural of extremum), of the function on the interval. The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum. They are also sometimes called the global minimum and global maximum.

4. Definition of Extreme Values on an Interval • The definition given is slightly different than the one given in your textbook • Most of the time we are only concerned with extreme values in an interval, rather than on the entire domain (many functions have neither absolute maxima nor absolute minima over the entire domain) • Over an interval, extrema can occur either in the interior or at the endpoints • It is possible for a function to have no extrema on an interval

5. Definition of Extreme Values on an Interval

6. Definition of Extreme Values on an Interval

7. Definition of Extreme Values on an Interval

8. Example 1: Exploring Extreme Values Use the graphs of and on to determine absolute maxima and minima (if any).

9. Example 1: Exploring Extreme Values

10. The Extreme Value Theorem THEOREM: If is continuous on a closed interval , then has both a maximum value and a minimum value on the interval.

11. The Extreme Value Theorem • The proof of this theorem requires more advanced calculus, so we will take the theorem as given • Note that continuity is a requirement of the proof; if we know that the function is not continuous on a given interval, then we cannot use the theorem • In plain words, this tells us that a function is guaranteed to have both a maximum and a minimum value on a closed interval (if continuous) • These extrema may be either at the endpoints of the interval or the interior of the interval

12. The Extreme Value Theorem

13. The Extreme Value Theorem

14. The Extreme Value Theorem

15. The Extreme Value Theorem

16. Local (Relative) Extrema • In addition to absolute (or global) extrema, which are always the greatest/least function value on an interval, we will want to define relative (local) extrema • These occur when “nearby” values are all less (for a relative maximum) or greater (for a relative minimum) • Relative extrema may also be absolute extrema, but not all relative extrema are absolute extrema (but all absolute extrema are also relative extrema)

17. Local (Relative) Extrema DEFINITION: Let be an interior point of the domain of the function . Then is a • local maximum value at if and only if for all in some open interval containing • local minimum value at if and only if for all in some open interval containing • A function has a local maximum or local minimum at an endpoint if the appropriate inequality holds for all in some half-open interval containing

18. Local (Relative) Extrema

19. Local (Relative) Extrema • Relative extrema in the interior of an interval occur at points where the graph of a function changes direction (from increasing to decreasing, or vice versa) • We would like to be able to find both absolute and relative extrema for a function over a closed interval • To narrow down the possibilities for the interior of an interval, we can ask, “is there anything about relative extrema by which we can identity them?”

20. Critical Point DEFINITION: Let be a function defined over some interval . A point , where is in the interior of , at which or is not differentiable is called a critical point of . The number in the interval is called a critical number of .

21. Critical Point • Note that critical points occur where either • is zero • is not differentiable • The next theorem (not in your text and presented without proof) gives us an answer to our previous question

22. Relative Extrema Occur Only at Critical Numbers THEOREM: If has a relative minimum or relative maximum at , then is a critical number of (and is a critical point of ).

23. Relative Extrema Occur Only at Critical Numbers • This theorem joins the definition of a critical number with the definition of relative extrema • Specifically, it says that relative extrema occur only at critical numbers • Therefore, to find relative extrema, we need only find critical numbers, which occur where either is zero or where is not differentiable • However, we must be clear about what it does not say: it does not say that, if we find a critical number, then we have found a relative extremum

24. Relative Extrema Occur Only at Critical Numbers

25. Finding Absolute Extrema • We now have enough understanding to be able to find the absolute extrema on a closed interval • If we have a closed interval, the Extreme Value Theorem guarantees that there exist both an absolute maximum and absolute minimum in the interval • These absolute extrema may occur at either the endpoints or in the interior • If absolute extrema occur in the interior, then they occur at relative extrema (which, in turn, occur at critical numbers) • To find absolute extrema, we will first find relative extrema in the interior (i.e., find critical numbers), evaluate the function at this

26. Finding Absolute Extrema • To find absolute extrema • We will first find relative extrema in the interior (i.e., find critical numbers) • Evaluate the function at all critical numbers • Evaluate the function at its endpoints • Compare these values; the largest of these is the absolute maximum and the smallest is the absolute minimum

27. Example 3: Finding Absolute Extrema Find the absolute maximum and minimum values of on the interval .

28. Example 3: Finding Absolute Extrema Find the absolute maximum and minimum values of on the interval . Find the critical numbers by taking the derivative: Critical numbers occur where is zero or where is not differentiable. Note that is never zero. However, is not defined at , which means that is not differentiable at , by our definition, is a critical number (and is a critical point). Now, check the function values at , , and : The maximum is approximately 2.08 and occurs at ; the minimum is 0 and occurs at

29. Example 3: Finding Absolute Extrema

30. Example 4: Finding Extreme Values Find the extreme values of .

31. Example 4: Finding Extreme Values Find the extreme values of . First note that no interval is provided. From the function it is clear that the domain is . Since this is not a closed interval, we cannot conclude that the function has both an absolute minimum and absolute maximum. We differentiate to find critical numbers: We have that if and this is the only critical number because critical numbers are only interior values of an interval. The critical point occurs at

32. Example 4: Finding Extreme Values Find the extreme values of . How can we know whether is a maximum, minimum, or neither? If , then the denominator of decreases, so its reciprocal increases. The same is true if . This means that, for all , and by definition this means that we have an absolute minimum at . Is there an absolute maximum? As approaches 2 from the left, the denominator approaches zero, so the function approaches infinity. The same is true as approaches from the right. So this function has no maximum value.

33. Example 4: Finding Extreme Values

34. Example 5: Finding Local Extrema Find the local extrema of

35. Example 5: Finding Local Extrema Find the local extrema of This function cannot have absolute extrema since function values continue increasing to the right of 1, and continue decreasing to the left of 1. What about local (relative) extrema? We differentiate by differentiating the two pieces of the function. But we must determine whether the function is differentiable at . To do this, find the left- and right-hand derivatives:

36. Example 5: Finding Local Extrema Find the local extrema of The derivatives are different, so is not differentiable at .

37. Example 5: Finding Local Extrema Find the local extrema of So our derivative is Therefore we have critical points at and . Since the function piece defined for is a parabola that opens down, then must be a local maximum.

38. Example 5: Finding Local Extrema We can determine how to classify the critical point by examining what happens to the function values at on either side of (but near) . To the left of , note that , , ; in general, these nearby values are greater than 3. To the right of : , , ; in general, these nearby values are greater than 3. So we have, that for values near so by definition, occurs at a local minimum.

39. Example 5: Finding Local Extrema

40. Example 6: Finding Local Extrema Find the local extrema of .

41. Example 6: Finding Local Extrema Find the local extrema of . The textbook asks you to use a calculator, but differentiating this function is not beyond your ability if you first note the following: Suppose that is a differentiable function of . Then can be written as a piecewise defined function Then we get if and , if . We can ignore the absolute value sign!

42. Example 6: Finding Local Extrema Find the local extrema of . Now, if , then Find (with respect to ): Therefore,

43. Example 6: Finding Local Extrema Find the local extrema of . Find where is zero and where is not differentiable. The function is not differentiable at , but this doesn’t count because is not defined (so zero is not in the domain of ). We have . The function values for these critical numbers are

44. Example 6: Finding Local Extrema Find the local extrema of . Finally, check some nearby function values to determine whether these are local maxima, minima, or neither. At : , ; this appears to be a local maximum At : , ; this, too, appears to be a local maximum.

45. Example 6: Finding Local Extrema

46. Exercise 5.1