MAT 135 – CalculusActivity for 4.1: Minimum and Maximum Values

Review: A function f has an absolute maximum (minimum) value 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 localmaximum (minimum) value if it is the largest (smallest) value in its local neighborhood, not necessarily thelargest (smallest) value on the entire domain of f .

1 How can local and absolute extreme values combine in a singlegraph?

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 eachof these five coefficients; do so by going to the Anim menu, then to Individual, and choose A toopen a window with a slider that sets the value for the coefficient A. Repeat this for B, C, D, andE. You can now play with the sliders to change the values of the coefficients and watch the curvechange shape as you do so.

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

http://screencast.com/t/L3yVNWuCxd8

Using the sliders, come up in each item below with a function having the description given. In eachcase, once you have come up with an appropriate function, record your values of A, B, C, D, andE 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 localminimum values is also the absolute minimum value but the local maximum value is not theabsolute maximum value

3. A function having a local minimum value and a local maximum value, where one of the localmaximum values is also the absolute maximum value but the local minimum value is not theabsolute 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 thisactivity.

1

Table for Recording Coefficient Values

Question A B C D E

1

2

3

4

5

6

2 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 extremevalues. Go back to each of those functions and estimate the value of the derivative at eachlocal extreme value. You should be able to do this by quick visual inspection and little to nocalculations. What appears to be true about the derivative value of your function at everyextreme value?

2. Now go to Winplot and graph y = |x2−1|. The Winplot syntax for this would be abs(x^2-1).Notice the cusps (sharp turning points) on the graph at x = −1 and x = 1. Explain why weshould consider the graph to have a local minimum value at these points. (In fact, these arealso the absolute minimum values of the graph.) What can you say about the derivative atthese 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.Then f ′(c), the derivative value at x = c, either or .

4. Your discovery in question 3 will be a heavily used tool for locating extreme values of afunction, but we must use it with some caution. On Winplot, plot the graph of y = x3 + 1and look at the graph when x = 0. Use a slider, visual inspection, or algebra to get the valueof the derivative dy/dx at x = 0. Is it true that every time the derivative of a function equals0, 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 eitherf ′(c) = 0 or f ′(c) does not exist.

For example, x = 0 is a critical number for y = x3 + 1, and the function y = |x2 − 1| has threecritical numbers: x = 1, x = −1, and x = 0. At the first two, the derivative is undefined (becauseof the cusps). At the third, the derivative is 0 (because the tangent line is horizontal).

1. Consider the graph of f , below:

2

-1 1 2 3 4 5 6 7 8

6

-2

-1

1

2

3

4

5

X Axis

Y Ax

is

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 numbersyou found. Does every critical number yield a local extreme value on f? Are there anylocal extreme values of f which do not occur at critical numbers?

3. Repeat the previous question with the function f(x) = xe−x. (This has just one criticalnumber.) You may use Wolfram|Alpha to do mechanical calculations, but be advised thatyou should be able to do this by hand for quizzes and Assessments.

4. Below are two schematic diagrams that attempt to show the relationship between criticalnumbers of a function f and local extreme values of f . The one on the left claims that everylocal extreme value happens at a critical number. The second one says that every criticalnumber gives a local extreme value. Which one is correct? Are both correct?

Local extreme

values of f

Critical numbers of f

Critical numbers of

f

Local extreme values of f

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

4 When must a function have absolute extreme values?

1. On Winplot, plot y = x3 + 1. Note that it has no absolute extreme values or local extremevalues at all. However, if we change the domain of this function from (−∞,∞) to somethingelse, things may change. Let’s redefine the domain of this function to the closed interval

3

[−2, 3] (that is, all x values between −2 and 3 and including both endpoints x = −2 andx = 3). On Winplot, we can do this by going to the Inventory screen, checking the “LockInterval” box, and then entering −2 for “low x” and 3 for “high x”. This will graph y = x3+1only 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 anabsolute minimum and an absolute maximum on this interval?

4. Now create a new plot in Winplot an enter in a fifth-degree polynomial of your choosing.Note that no matter what coefficients you choose, the graph will have no absolute minimumand no absolute maximum. (Why is that?) But then restrict the domain to a closed interval.Do you get an absolute minimum now? What about an absolute maximum?

5. Based on your 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 anabsolute maximum value and an absolute minimum value.

And you’d almost be right. However, consider the following function:

y =

{1

x−1 if x 6= 1

2 if x = 1

Draw th graph of this function (by hand, or use Winplot if you recall how to plot piecewisefunctions) only on the closed interval [−2, 2]. If the statement above were true, then thisfunction ought to have an absolute maximum value and an absolute minimum value on[−2, 2], since the function is defined at every point in the closed interval [−2, 2]. (In fact,this function’s domain is R, the entire real number line.) But this is not the case – thisfunction has neither an absolute minimum value nor an absolute maximum value on thisclosed interval! Why not? Based on this example, fill in the blank below to make a statementthat always works:

If f is a function defined only over a closed interval [a, b] and f isat all points in [a, b], then f has both an absolute maximum value and an absoluteminimum value.

5 Wrapping Up

Review your group’s work on this activity and answer the following questions. These answers, alongwith your verbal contributions to the debriefing session and a follow-up exercise, will be used foryour attendance grade.

1. What did you learn about local extreme values and absolute extreme values of functions inthe first part of this activity?

2. What can we say about the derivative value of a function at any place where it has a localextreme value?

4

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 extremevalue at x = c? If not, give an example.

5. Under what conditions must a function have both an absolute minimum and an absolutemaximum?

This work is licensed under the Creative Commons Attribution-Share Alike 3.0 United States License. To

view a copy of this license, visit http://creativecommons.org/licenses/by-sa/3.0/us/ or send a letter

to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

5