Extreme values: Maxima and minima. October 16, / 12

Transcription

1 Extreme values: Maxima and minima October 16, / 12

2 Motivation for Maxima and Minima Imagine you have some model which predicts profits / cost of doing something /.... Then you probably want to find a way of deciding what choices result in the largest (for profit) or smallest (for cost) result possible. Today s Goal: Can calculus help us here? October 16, / 12

3 Motivation for Maxima and Minima Imagine you have some model which predicts profits / cost of doing something /.... Then you probably want to find a way of deciding what choices result in the largest (for profit) or smallest (for cost) result possible. Today s Goal: Can calculus help us here? Today s outline: (1)Before we want to find best possible choices, we will want to have a reason to beleive that there is a best possible choice. (2) Then we will use calculus to reduce the finding of the best choice (or worst choice) to solving some equation involving the derivative! October 16, / 12

4 Hiking Imagine you are walking and your height as a function of time is given by h(t). Suppose that the graph is given by : October 16, / 12

5 Hiking Imagine you are walking and your height as a function of time is given by h(t). Suppose that the graph is given by : Where does it look like you are at a peak? Where does it look like you are in a valley? Where on your hike are you at the greatest elevation? Where are you at your lowest elevation? October 16, / 12

6 Hiking Imagine you are walking and your height as a function of time is given by h(t). Suppose that the graph is given by : Where does it look like you are at a peak? Where does it look like you are in a valley? Where on your hike are you at the greatest elevation? Where are you at your lowest elevation? If this were a graph of expected price of a doodad over time, then when is the best time to buy a Thneed? October 16, / 12

7 Definitions: Global maxima and minima For a function f, defined on an interval I containing a point c. c is called an Absolute Maximum (or Global Maximum) if f (c) is greater and or equal to f (x) for all values of x in I. For a function f, defined on an interval I containing a point c. c is called an Absolute Minimum (or Global Minimum) if f (c) is greater and or equal to f (x) for all values of x in I. October 16, / 12

8 Definitions: Global maxima and minima For a function f, defined on an interval I containing a point c. c is called an Absolute Maximum (or Global Maximum) if f (c) is greater and or equal to f (x) for all values of x in I. For a function f, defined on an interval I containing a point c. c is called an Absolute Minimum (or Global Minimum) if f (c) is greater and or equal to f (x) for all values of x in I. Where are the absolute maxima and minima for these functions. How many are there? cos(x) (x 1) 2 1/x This function on (0, 2π) on [0, 2] on (0, ) on [0, 2] October 16, / 12

9 Definitions: Global maxima and minima For a function f, defined on an interval I containing a point c. c is called an Absolute Maximum (or Global Maximum) if f (c) is greater and or equal to f (x) for all values of x in I. For a function f, defined on an interval I containing a point c. c is called an Absolute Minimum (or Global Minimum) if f (c) is greater and or equal to f (x) for all values of x in I. Where are the absolute maxima and minima for these functions. How many are there? cos(x) (x 1) 2 1/x This function on (0, 2π) on [0, 2] on (0, ) on [0, 2] That is weird. Not all functions have an absolute maximum / minimum. And some have many maxima / minima. October 16, / 12

10 The extreme value theorem What went wrong in these examples? October 16, / 12

11 The extreme value theorem What went wrong in these examples? Well some of the functions just weren t continuous. Discontinuous functions can do anything they want. October 16, / 12

12 The extreme value theorem What went wrong in these examples? Well some of the functions just weren t continuous. Discontinuous functions can do anything they want. In some examples the functions were defined on intervals that were lacking an endpoint (or the endpoint was infinite.) The function could keep increasing (or decreasing) as it approached that endpoint. October 16, / 12

13 The extreme value theorem What went wrong in these examples? Well some of the functions just weren t continuous. Discontinuous functions can do anything they want. In some examples the functions were defined on intervals that were lacking an endpoint (or the endpoint was infinite.) The function could keep increasing (or decreasing) as it approached that endpoint. If we rule out these problems then all functions have (possibly many) maxima / minima. Theorem (The extreme value theorem) If a function is continuous on some closed interval, then it has an absolute maximum and an absolute minimum on that closed interval. October 16, / 12

14 The extremal value theorem Theorem (The extremal value theorem) If a function is continuous on some closed, bounded interval [a, b], then it has an absolute maximum and an absolute minimum on that closed interval. What this theorem tells us is that looking for absolute maxima and minima is a reasonable thing to do, as long as we are working on closed intervals. If we are working on an open internal, then there are no guarantees. October 16, / 12

15 The extremal value theorem Theorem (The extremal value theorem) If a function is continuous on some closed, bounded interval [a, b], then it has an absolute maximum and an absolute minimum on that closed interval. What this theorem tells us is that looking for absolute maxima and minima is a reasonable thing to do, as long as we are working on closed intervals. If we are working on an open internal, then there are no guarantees. Does f (x) = x + cos(x) have a absolute maximum on [ π, π]? How about an absolute minimum? October 16, / 12

16 The extremal value theorem Theorem (The extremal value theorem) If a function is continuous on some closed, bounded interval [a, b], then it has an absolute maximum and an absolute minimum on that closed interval. What this theorem tells us is that looking for absolute maxima and minima is a reasonable thing to do, as long as we are working on closed intervals. If we are working on an open internal, then there are no guarantees. Does f (x) = x + cos(x) have a absolute maximum on [ π, π]? How about an absolute minimum? Does is have an absolute maximum on (, )? What about an absolute Minimum? Try graphing October 16, / 12

17 Today s goal: Finding absolute maxima and minima, now that we know they exist. We know that continuous functions on closed intervals have global maxima. Can we find them? October 16, / 12

18 Today s goal: Finding absolute maxima and minima, now that we know they exist. We know that continuous functions on closed intervals have global maxima. Can we find them? Calculus studies things locally. Think about the definition of the derivative for example. Motivated by this we will localize the idea of a maximum. October 16, / 12

19 Today s goal: Finding absolute maxima and minima, now that we know they exist. We know that continuous functions on closed intervals have global maxima. Can we find them? Calculus studies things locally. Think about the definition of the derivative for example. Motivated by this we will localize the idea of a maximum. If c is a global maximum then f (c) is greater than or equal to f (x) for x in I. Then in particular f (c) is greater than or equal to f (x) for x close to c in I. October 16, / 12

20 Today s goal: Finding absolute maxima and minima, now that we know they exist. We know that continuous functions on closed intervals have global maxima. Can we find them? Calculus studies things locally. Think about the definition of the derivative for example. Motivated by this we will localize the idea of a maximum. If c is a global maximum then f (c) is greater than or equal to f (x) for x in I. Then in particular f (c) is greater than or equal to f (x) for x close to c in I. Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. If c is a global maximum, then it is a local maximum. So If you find a list containing every local maximum then that list contains the global maximum? We have some plots to look at on the notes. How can calculus help us here? October 16, / 12

21 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? October 16, / 12

22 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: October 16, / 12

23 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum October 16, / 12

24 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum Similarly if f (c) < 0 then c is not a local maximum. Taking a step backwards would produce greater height. October 16, / 12

25 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum Similarly if f (c) < 0 then c is not a local maximum. Taking a step backwards would produce greater height. Theorem If c is a local maximum of f then either or. Example: Find a list of candidates to be local maxima of x 3 3x + 2 on [ 1, 2] or October 16, / 12

26 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum Similarly if f (c) < 0 then c is not a local maximum. Taking a step backwards would produce greater height. Theorem If c is a local maximum of f then either f (c) = 0 or or. Example: Find a list of candidates to be local maxima of x 3 3x + 2 on [ 1, 2] October 16, / 12

27 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum Similarly if f (c) < 0 then c is not a local maximum. Taking a step backwards would produce greater height. Theorem If c is a local maximum of f then either f (c) = 0 or f (c) does not exist or. Example: Find a list of candidates to be local maxima of x 3 3x + 2 on [ 1, 2] October 16, / 12

28 Finding local maxima and minima Definition For a function f, defined on an interval I containing a point c. c is called a local Maximum if f (c) is greater and or equal to f (x) for all x nearby c in I. How can calculus help us here? Think about what the derivative means: If f (c) > 0 then you are still ascending at c. If you take one more step then you will be higher. c does not produce a local maximum Similarly if f (c) < 0 then c is not a local maximum. Taking a step backwards would produce greater height. Theorem If c is a local maximum of f then either f (c) = 0 or f (c) does not exist or c is an endpoint of the interval. Example: Find a list of candidates to be local maxima of x 3 3x + 2 on [ 1, 2] October 16, / 12

29 Critical points Definition (Critical points) For a function f defined on an interval I c is a critical point if f (c) = 0 or f (c) does not exist or c is an endpoint of the interval Notice that the points on the boundary of the interval are automatically critical points. The theorem on the previous slide is Theorem If c is a local maximum of f then c is a critical point. Example: What are the critical points of x 3 x 2 x 1 on [ 1, 2]. Find the global maximum. Find the global minimum. October 16, / 12

30 A general strategy: An Algorithm to find the global maximum of f (x) on the interval I : (1) Compute f (x) (2) Find the critical points of f (INCLUDING THE BOUNDARY POINTS) (3) Evaluate the function at those points and see which one is the biggest. Example: Find the global maximum of f (x) = x 3 x on [ 10, 10]. Example: Find the global maximum of f (x) = x 4 x on [ 10, 10]. October 16, / 12

31 maximizing areas of rectangles Of all rectangles which have total perimeter 50, which one has the greatest area? Strategy: 1 In terms of length (l) and width (w), what does it mean to have perimeter 50? Solve for l in terms of w. (or w in terms of l.) 2 Express area in terms of l and w. Eliminate one of these variable. 3 Since the perimeter is 50, could l possible be greater than 50? Do negative numbers make sense for l and w? 4 Write down the closed interval which is the domain of interest. 5 Maximize. October 16, / 12

32 Group Work Maximize x 3 x + 1 on [ 10, 10] Maximize f (x) = x on [ 10, 10]. x + 1 Maximize x + cos 1 (x) on [ 1, 1] October 16, / 12