Exploring Minimum And Maximum Values

MAT 135 – Calculus Activity for 4.1: Minimum and Maximum Values Review: In the outside reading, you learned about local and absolute maximum values of a function and the differences between them. A function f has an absolute maximum (minimum) at x = c if f (c) is the largest (smallest) value that the function will ever take on the domain we are working on. By contrast, a function has a local maximum (minimum) value if it is the largest (smallest) value in its local neighborhood, not necessarily the largest (smallest) value on the entire domain of f . Guidelines for this activity: In this activity you will discover all the important calculus properties of local and absolute extreme values on your own. There will be no lecturing by the professor. This is being done to help make you an independent learner and practitioner of mathematics who is capable of assimilating technical information and solving problems on your own without needing an expert guiding your steps. To that end, my role in this activity will be (1) to answer questions intended to clarify instructions on the activity or with the technology, (2) to make sure all groups are progressing at a reasonable pace through the activity, and (3) to help you ask proactive questions of your own work if you get stuck. I will not answer any question of the form “Is this right?” or a variation thereof. You will have to check your own work and be confident of it. When the activity is over – we plan on stopping around 7:20 – we will pool our results and discuss what happened, and correct mathematics ought to emerge from that discussion.

1

How can local and absolute extreme values combine in a single graph?

Start up Winplot, and enter in the following function to plot: A*x^4 + B*x^3 + C*x^2 + D*x + E This will be a fourth-degree polynomial with five coefficients. You can open up a slider for each of these five coefficients; do so by going to the Anim menu, then to Individual, and choose A to open a window with a slider that sets the value for the coefficient A. Repeat this for B, C, D, and E. You can now play with the sliders to change the values of the coefficients and watch the curve change shape as you do so. For example, the window below shows where I’ve set A = 2, B = −5, C = −2, D = 2, and E = 3 so that the graph is that of y = 2x4 − 5x3 − 2x2 + 2x + 3:

If you need further guidance, a brief video of this process is at 1

http://screencast.com/t/L3yVNWuCxd8 Using the sliders, come up in each item below with a function having the description given. In each case, once you have come up with an appropriate function, record your values of A, B, C, D, and E in the table provided. 1. A function having at least one local minimum value and at least one local maximum value, but no absolute extreme values 2. A function having a local minimum value and a local maximum value, where one of the local minimum values is also the absolute minimum value but the local maximum value is not the absolute maximum value 3. A function having a local minimum value and a local maximum value, where one of the local maximum values is also the absolute maximum value but the local minimum value is not the absolute minimum value 4. A function with at least one local minimum value but no local or absolute maximum values 5. A function with at least one local maximum value but no local or absolute minimum values 6. A function with no local or absolute extreme values at all Leave Winplot open and do not delete your plot, because you’ll be using it in the next parts of this activity. Table for Recording Coefficient Values Question 1 2 3 4 5 6

2

A

B

C

D

E

What do derivatives have to say about extreme values? 1. In items 1–5 in Part 1 of the activity, you came up with functions that had local extreme values. Go back to each of those functions (you can just reset the sliders to do so) and find the value of the derivative of your function at each local extreme value. You can do this by quick visual inspection; or you can go to the One menu, then open a slider and click the box that displays the tangent line on the graph, then slide the cursor to the extreme value and read off the slope. What appears to be true about the derivative value of your function at every extreme value? 2. Now create a new plot (without closing out your first one) and graph y = |x2 − 1|. The Winplot syntax for this would be abs(x^2-1) 2

Notice the cusps at x = −1 and x = 1. Explain why we should consider the graph to have a local minimum value at these points. (In fact, these are also the absolute minimum values of the graph.) What can you say about the derivative at these two points? 3. Put the information from questions 1 and 2 together to fill in the blanks below: Suppose f is a function that has a local minimum or local maximum value at x = c. or . Then f 0 (c), the derivative value at x = c, either 4. Your discovery in question 3 will be a heavily used tool for locating extreme values of a function, but we must use it with some caution. On Winplot, plot the graph of y = x3 + 1 and look at the graph when x = 0. Use a slider, visual inspection, or algebra to get the value of the derivative dy/dx at x = 0. Is it true that every time the derivative of a function equals 0, that we obtain a local extreme value there?

3

Critical information

Based on your discovery in the previous part of this activity, we will define the following idea: Let f be a function. We say that a number c is a critical number of f if either f 0 (c) = 0 or f 0 (c) does not exist. For example, x = 0 is a critical number for y = x3 − 1. The function y = |x2 − 1| has three critical numbers: x = 1, x = −1, and x = 0. At the first two, the derivative is undefined (because of the cusps). At the third, the derivative is 0. 1. Consider the graph of f , below: 6

5

4

Y Axis

3

2

1

-1

1

2

3

4

5

6

7

8

-1

-2 X Axis

State the critical numbers of f and explain what makes each one a critical number. 2. Consider the function f (x) = x4 − 4x3 . (a) Find all the critical numbers of f . (There are two of them.) (b) Plot f on Winplot in an appropriate viewing window and look at the critical numbers you found. Does every critical number yield a local extreme value on f ? Are there any local extreme values of f which do not occur at critical numbers? 3

3. Repeat the previous question with the function f (x) = xe−x . (This has just one critical number.) 4. Below are two schematic diagrams that attempt to show the relationship between critical numbers of a function f and local extreme values of f . The one on the left claims that every local extreme value happens at a critical number. The second one says that every critical number gives a local extreme value. Which one is correct? Are both correct?

Critical numbers of f

Local extreme values of f

Local extreme values of f

Critical numbers of f

5. If you know that f has a critical number at x = c, what are some ways to tell whether it is a local minimum, local maximum, or neither one?

4

When must a function have absolute extreme values? 1. Open up a new graph in Winplot and plot y = x3 + 1. Note that it has no absolute extreme values or local extreme values at all. However, if we change the domain of this function from (−∞, ∞) to something else, things may change. Let’s redefine the domain of this function to the closed interval [−2, 3] (that is, all x values between −2 and 3 and including both endpoints x = −2 and x = 3). On Winplot, we can do this by going to the Inventory screen, checking the “Lock Interval” box, and then entering −2 for “low x” and 3 for “high x”. This will graph y = x3 + 1 only over the closed interval [−2, 3], so it looks like just a piece of the former graph. 2. Examine y = x3 + 1 on the interval [−2, 3]. Does it have an absolute maximum value now? Where is it? Does it have an absolute minimum now? Where is it? 3. Change the interval from [−2, 3] to something else of your choosing. Does y = x3 + 1 have an absolute minimum and an absolute maximum on this interval? 4. Go back to your graph from question 1 in the first part of this activity, where you made a fourth-degree polynomial having some local extrema but no absolute extrema. Change the domain to any closed interval you like. Does your function now have an absolute minimum and an absolute maximum on this interval? 5. Repeat for questions 4 and 5 from the first part of this activity (where you came up with more functions having no absolute extreme values). 6. Repeat again for question 2, where you came up with a function having an absolute minimum value but no absolute maximum value. Where is the absolute minimum? Where is the (new) absolute maximum? 4

7. Based on your graphing work above, you might begin to think that the following must be true: If f is a function defined only over a closed interval [a, b], then f has both an absolute maximum value and an absolute minimum value. And you’d almost be right. However, go to Winplot and graph y = 1/x only on the closed interval [−2, 2]. Does it have an absolute minimum? An absolute maximum? Why not? Based on this example, fill in the blank below to make a statement that always works: If f is a function defined only over a closed interval [a, b] and f is at all points in [a, b], then f has both an absolute maximum value and an absolute minimum value.

5

Wrapping Up

Review your group’s work on this activity and answer the following questions. These answers, along with your verbal contributions to the debriefing session and a follow-up exercise, will be used for your in-class group work grade tonight. 1. What did you learn about local extreme values and absolute extreme values of functions in the first part of this activity? 2. What can we say about the derivative value of a function at any place where it has a local extreme value? 3. What is a critical number ? 4. Is it true that, if a function has a critical number at x = c, then it must have a local extreme value at x = c? If not, give an example. 5. Under what conditions must a function have both an absolute minimum and an absolute maximum?

5