1.2: Minimums and Maximums
Learning objectives
- Analyze a situation and determine if it is maximization or minimization
- Write an expression to represent such a situation
- Use a graphing utility to find a maximum or minimum
Introduction
Consider the following example: you run a business making boxes of cookies. While you earn money by selling cookies, it costs you money to construct every box. (These costs include the materials, as well as all of the costs to run the business, such as paying your employees!)
As the owner of the business, you will want to maximize your profits, and you will want to minimize your costs. That is, you want to make the most money that you can, while spending as little as you can in the process. In this chapter you will examine situations in which you will want to maximize or minimize some quantity. You will learn how to identify maxima and minima by writing functions to represent situations, and by examining the graphs of these functions.
Min-Max situations
In the example in the introduction, one quantity would be maximized, while the other would be minimized. In general, however, not every situation would necessarily include both of these ideas. For example, consider a situation in which you want to make a box that has a volume of 12 cm^{3}, but you want the surface area of the box to be as small as possible.
In this situation, you want to find the minimum surface area, given a fixed volume.
You might also have a situation in which you are given a fixed amount of material with which to make the box, and you want to make as large a box as possible. In this situation, you would want to find the maximum volume of the box, given a fixed surface area.
Whether you will be looking for a maximum or a minimum depends on the specific situation. Consider the situations in example 1.
Example 1: In each situation determine if a quantity should be maximized or minimized.
- a. You have 100 feet of fence to enclose a field, and you want to create the largest field possible.
- b. You run a factory that packages toilet paper, and you want to use the least amount of plastic possible for each roll.
Solution:
- a. This situation involves maximizing the area of the field.
- b. This situation involves minimizing the amount of plastic used per roll. (This would be the surface area of a cylinder.)
Expressions to be minimized or maximized
Consider the example above of the box with volume 12 cm^{3}. The volume of a box is the product of its three dimensions: length, width, and height. So we can express the volume of the box as an equation: LWH = 12. There are infinitely many boxes with volume 12. Three possible boxes are shown below:
We can find the surface area of each box by adding up the area of the six faces. The first box has four faces with area 6 and 2 faces with area 4, so the surface area of the box is 32 cm^{2}. Similarly, you can find that the second box has surface area 50cm^{2}. Finally the surface area of the third box is 44cm^{2}. Therefore, of these three boxes, the first one has the smallest surface area. However, this box does not necessarily have the smallest surface area possible for a box with the given volume. To find the box with minimum surface area requires that we know a bit more information about the box.
Let’s consider a slightly simpler situation: a box with a square base and a fixed volume of 12 cm^{3}. Now let the length and the width of the box be x cm, and the height be h cm. We can write the volume equation as \begin{align*}x \cdot x \cdot h = x^{2}h=12\end{align*}. We can also express the surface area in terms of x and h:
- \begin{align*}\,\! \text{Surface area} = S = 4xh+2x^{2}\end{align*}
(Make sure you understand this formula: The base and the top are squares with area = \begin{align*}x^{2}\end{align*} and the four sides are each rectangles of area equal to \begin{align*}xh\end{align*} ).
We can express the surface area as a function of x if we consider the volume equation and the surface area equation as a system of equations:
- \begin{align*}\begin{cases} x^2h = 12 \\ 4xh + 2x^2 = S \\ \end{cases}\end{align*}
We want to work with the surface area equation since that is what we want to minimize. However, there are currently three variables in that equation: S, x, and h. It will be easier to graph and analyze surface area if we can express S in terms of just one other variable. So, we want to use substitution to get rid of one of the variables. We can use the volume equation to rewrite the surface area equation as a function of x.
First, rewrite the volume equation:
\begin{align*}x^2h = 12 \Rightarrow h = \frac{12} {x^2}\end{align*}
Now, use substitution:
\begin{align*} S(x)&=4xh+2x^{2}\\ &=4x \left( \frac{12}{x^2}\right) +2x^{2}\\ &=\frac{48}{x}+2x^{2} \end{align*}
The values of the function S(x) represent different possibilities for the surface area of the box, given that the base is a square, and given that the volume of the box is 12 cm^{3}. Therefore to identify the minimum surface area, we need to find the lowest function values for S(x). When you study calculus, you will learn algebraic techniques for determining maximum and minimum values of a function. Here, we will rely on our knowledge of graphing to identify the minimum value of S(x).
The graph below shows the function S(x) on the interval [0,5]. It is important to note that the function S(x) has the domain of all real numbers except zero, but we are only interested in x values greater than zero since x represents the side length of the box.
By examining the graph, we can see that the lowest point is between x = 2 and x = 3. If you use a “minimum” function on a graphing utility, you will find that the minimum point is approximately (2.3, 31.4). This tells us that when the side length of the box is approximately 2.3 cm, the surface area is approximately 31.4 cm^{2}, which is the smallest it can be.
- The example below asks you to find a maximum value.
Example 2: You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You want the enclosed land to be a rectangle. What size rectangle should you make with the fence in order to maximize the area of the rectangular enclosure?
Solution: The plot of land will look like the picture below:
The area of the rectangular plot is the product of its length and width. We can write the area as a function of x: A(x) = xh. We can eliminate h from the equation if we consider that we have 100 feet of fence, and we write an equation about how we are using that 100 feet of fence: x + 2h = 100. (The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation for h and substitute into the area equation:
- \begin{align*} & x+2h=100 \\ \Rightarrow & 2h=100-x \\ \Rightarrow & h=50-\frac{x}{2} \end{align*}
\begin{align*} A(x) &= xh \\ &= x \left( 50 - \frac{x} {2} \right) \\ &= 50x-\frac{x^2}{2} \end{align*}
The graph of A(x) is shown here on the interval [0,100]. Using a maximum function on a graphing utility tells us that the point (50,1250) is the maximum point. This tells us that when the rectangle’s width is 50 ft, the area is 1250 ft^{2}.
In both of the previous two examples, we have considered a function on a subset of its domain. In particular, we limited our graphs to positive x values, as we were dealing with real quantities that could not reasonably be negative. However, if you examine the functions \begin{align*}S(x)=\frac{48}{x}+2x^{2}\end{align*} and \begin{align*}A(x)=50x-\frac{x^2}{2}\end{align*} for all real values of x, you will notice a difference in their graphs: the point we identified on A(x) is the highest point the function has on its entire domain. This is not the case for S(x). In the next lesson we will consider more functions like S(x). In the remainder of this lesson we will consider functions like A(x).
Symbolic definition of minimum
Consider the function \begin{align*} f(x)=x^{2}-5\end{align*}. If you graph this function, you should see that the point (0, -5) is the lowest point on the graph. Even if we greatly increase the interval on which we view the graph, this point will always be the lowest point.
This point is referred to as the minimum value of the function. This minimum value is also referred to as a global minimum, to emphasize the fact that it is the lowest point on the entire graph.
We formally define the minimum point of a function as follows: The point (c, f(c)) is the minimum value of a function if f(c) ≤ f(a) for all elements a (a ≠ c) of the domain of f. Simply put: the point (c, f(c)) is the minimum if all other function values are greater than or equal to f(c).
- We can define the maximum of a function in a similar manner.
Symbolic definition of maximum
The function \begin{align*}A(x)=50x-\frac{x^2}{2}\end{align*} has a global maximum at the point (50, 1250). Formally we can define the maximum of a function as follows: The point (c, f(c)) is the maximum of f(x) if f(c) ≥ f(a) for all elements a (a ≠ c) of the domain of f. Simply put: the point (c, f(c)) is the maximum if all other function values are less than or equal to f(c). Together, maximum and minimum points are called extrema. It is important to note that not all functions have extrema.
Example 4: Determine if each function has a minimum or a maximum point
a. y = 2x - 1 | b. y = x^{4} |
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Solution
a. The graph of y = 2x - 1 is a line. It does not have a maximum or a minimum.
b. The graph of y = x^{4} has a minimum value at (0,0). It does not have a maximum.
Lesson Summary
In this lesson we have focused on the idea of extrema: the minimum and maximum values of a function. Throughout the lesson we have examined examples of situations that involved quantities to be maximized or minimized. For example, we found the minimum surface area of a box, given a fixed volume, and we found the maximum area of a rectangular enclose, given a fixed perimeter. In order to find extrema, it is necessary to consider the graphs of functions. It is important to note again that when you study calculus, you will learn algebraic methods for finding exact values of extrema. For now we have used graphs to approximate these values. It is also important to note that not all functions have global extrema. In the next lesson we will consider functions (like the surface area example in this lesson) that have extrema that are not “global”.
Points to Consider
- What information might you look for in a situation in order to determine if the quantity involved should be maximized or minimized?
- What do you think are the pros and cons of using a graphing utility to identify extrema?
- What aspects of the equations of functions might indicate that a function has a maximum or a minimum value?
Review Questions
- What quantity should be maximized? What quantity should be minimized? You are manufacturing chairs, and it costs you a certain amount of money to make each chair. You need to determine the selling price of the chairs.
- A rectangle has area 20 in^{2}. Write an expression for the perimeter of the rectangle as a function of its width x.
- What dimensions of the rectangle in problem #2 will minimize its perimeter? What is the minimum perimeter? (These values will be approximations.)
- In your own words, define the term “maximum of a function.”
- Explain how you can use a graph to identify global extrema of a function.
- A rectangle has a perimeter of 24 inches. What is the maximum area the rectangle can have?
- A cylindrical canister has a volume of 30 in^{3}. What is the radius of the canister with minimum surface area? (Volume of a cylinder is \begin{align*} V=\pi r^{2} h\end{align*}
- Consider the function f(x) = bx^{2} + 7. For what values of b will the function have a maximum?
- Consider the function \begin{align*}S(x)=\frac{48}{x}+2x^{2}\end{align*}. How can you tell that this function does not have a global maximum or minimum?
- A rectangle has perimeter P. Write a function for the area of the rectangle as a function of P and x, the width of the rectangle. What do you think will be the rectangle with maximum area?
Review Answers
- The cost per chair should be minimized. The profit (a function of the selling price) should be maximized.
- \begin{align*}P = 2x + \frac{40} {x}\end{align*}
- When x=4.472, the perimeter is about 17.889 feet.
- Answers will vary. Example: A maximum is the highest point on the graph, or the greatest y value of the function.
- Answers will vary. Example: After graphing a function, you need to look for the highest or lowest point on the graph. It is important to explore a large subset of the domain, unless you are very familiar with a particular kind of function.
- The rectangle with maximum area is a 6x6 square, with area 36 in^{2}.
- \begin{align*}r \approx 1.68\end{align*} inches.
- If b<0, the function will have a maximum.
- If you examine the graph for negative x values, you see that the graph has more than one “piece”. The piece that we did not consider in the example in the text goes above and below the point we identified in the example in the text.
- \begin{align*}A(x) = x \left (\frac{P} {2} - x\right)\end{align*} If you choose values of P, and examine the graphs, you will see that the rectangle with maximum area is always a square, like the one in problem 6.
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