Analysis of the Solution

Note that a decrease is expressed by a negative change or "negative increase." A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases.

The following video provides another example of how to find the average rate of change between two points from a table of values. https://www.youtube.com/watch?v=iJ_0nPUUlOg&feature=youtu.be The following video provides another example of finding the average rate of change of a function given a formula and an interval. https://www.youtube.com/watch?v=g93QEKJXeu4&feature=youtu.be

As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure 3 shows examples of increasing and decreasing intervals on a function.

Figure 3. The function [latex]f\left(x\right)={x}^{3}-12x[/latex] is increasing on [latex]\left(-\infty \text{,}-\text{2}\right){{\cup }^{\text{ }}}^{\text{ }}\left(2,\infty \right)[/latex] and is decreasing on [latex]\left(-2\text{,}2\right)[/latex].

This video further explains how to find where a function is increasing or decreasing. https://www.youtube.com/watch?v=78b4HOMVcKM

While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maximum. If a function has more than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a local minimum. The plural form is "local minima." Together, local maxima and minima are called local extrema, or local extreme values, of the function. (The singular form is "extremum.") Often, the term local is replaced by the term relative. In this text, we will use the term local.

Clearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.

For the function in Figure 4, the local maximum is 16, and it occurs at [latex]x=-2[/latex]. The local minimum is [latex]-16[/latex] and it occurs at [latex]x=2[/latex].

To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure 5 illustrates these ideas for a local maximum.

These observations lead us to a formal definition of local extrema.

Analyzing the Toolkit Functions for Increasing or Decreasing Intervals

We will now return to our toolkit functions and discuss their graphical behavior in the table below.

Function	Increasing/Decreasing	Example
Constant Function [latex]f\left(x\right)={c}[/latex]	Neither increasing nor decreasing
Identity Function [latex]f\left(x\right)={x}[/latex]	Increasing
Quadratic Function [latex]f\left(x\right)={x}^{2}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex] Decreasing on [latex]\left(-\infty,0\right)[/latex] Minimum at [latex]x=0[/latex]
Cubic Function [latex]f\left(x\right)={x}^{3}[/latex]	Increasing
Reciprocal [latex]f\left(x\right)=\frac{1}{x}[/latex]	Decreasing [latex]\left(-\infty,0\right)\cup\left(0,\infty\right)[/latex]
Reciprocal Squared [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex]	Increasing on [latex]\left(-\infty,0\right)[/latex] Decreasing on [latex]\left(0,\infty\right)[/latex]
Cube Root [latex-display]f\left(x\right)=\sqrt[3]{x}[/latex-display]	Increasing
Square Root [latex]f\left(x\right)=\sqrt{x}[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex]
Absolute Value [latex]f\left(x\right)=|x|[/latex]	Increasing on [latex]\left(0,\infty\right)[/latex] Decreasing on [latex]\left(-\infty,0\right)[/latex]

Use A Graph to Locate the Absolute Maximum and Absolute Minimum

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\text{-}[/latex] coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively.

To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure 10.

Figure 10

Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\left(x\right)={x}^{3}[/latex] is one such function.

A General Note: Absolute Maxima and Minima

The absolute maximum of [latex]f[/latex] at [latex]x=c[/latex] is [latex]f\left(c\right)[/latex] where [latex]f\left(c\right)\ge f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].

The absolute minimum of [latex]f[/latex] at [latex]x=d[/latex] is [latex]f\left(d\right)[/latex] where [latex]f\left(d\right)\le f\left(x\right)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex].

Example 10: Finding Absolute Maxima and Minima from a Graph

For the function [latex]f[/latex] shown in Figure 11, find all absolute maxima and minima.

Figure 11

Solution

Observe the graph of [latex]f[/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[/latex] and [latex]x=2[/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the y-coordinate at [latex]x=-2[/latex] and [latex]x=2[/latex], which is [latex]16[/latex].

The graph attains an absolute minimum at [latex]x=3[/latex], because it is the lowest point on the domain of the function’s graph. The absolute minimum is the y-coordinate at [latex]x=3[/latex], which is [latex]-10[/latex].

Key Equations

Average rate of change

[latex]\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]

Key Concepts

• A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data.
• Identifying points that mark the interval on a graph can be used to find the average rate of change.
• Comparing pairs of input and output values in a table can also be used to find the average rate of change.
• An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula.
• The average rate of change can sometimes be determined as an expression.
• A function is increasing where its rate of change is positive and decreasing where its rate of change is negative.
• A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
• A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
• Minima and maxima are also called extrema.
• We can find local extrema from a graph.
• The highest and lowest points on a graph indicate the maxima and minima.

Glossary

absolute maximum

the greatest value of a function over an interval

absolute minimum

the lowest value of a function over an interval

average rate of change

the difference in the output values of a function found for two values of the input divided by the difference between the inputs

decreasing function

a function is decreasing in some open interval if [latex]f\left(b\right)<f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]

increasing function

a function is increasing in some open interval if [latex]f\left(b\right)>f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]

local extrema

collectively, all of a function's local maxima and minima

local maximum

a value of the input where a function changes from increasing to decreasing as the input value increases.

local minimum

a value of the input where a function changes from decreasing to increasing as the input value increases.

rate of change

the change of an output quantity relative to the change of the input quantity

Can the average rate of change of a function be constant? 2. If a function [latex]f[/latex] is increasing on [latex]\left(a,b\right)[/latex] and decreasing on [latex]\left(b,c\right)[/latex], then what can be said about the local extremum of [latex]f[/latex] on [latex]\left(a,c\right)?[/latex] 3. How are the absolute maximum and minimum similar to and different from the local extrema? 4. How does the graph of the absolute value function compare to the graph of the quadratic function, [latex]y={x}^{2}[/latex], in terms of increasing and decreasing intervals? For exercises 5–15, find the average rate of change of each function on the interval specified for real numbers [latex]b[/latex] or [latex]h[/latex]. 5. [latex]f\left(x\right)=4{x}^{2}-7[/latex] on [latex]\left[1,\text{ }b\right][/latex] 6. [latex]g\left(x\right)=2{x}^{2}-9[/latex] on [latex]\left[4,\text{ }b\right][/latex] 7. [latex]p\left(x\right)=3x+4[/latex] on [latex]\left[2,\text{ }2+h\right][/latex] 8. [latex]k\left(x\right)=4x - 2[/latex] on [latex]\left[3,\text{ }3+h\right][/latex] 9. [latex]f\left(x\right)=2{x}^{2}+1[/latex] on [latex]\left[x,x+h\right][/latex] 10. [latex]g\left(x\right)=3{x}^{2}-2[/latex] on [latex]\left[x,x+h\right][/latex] 11. [latex]a\left(t\right)=\frac{1}{t+4}[/latex] on [latex]\left[9,9+h\right][/latex] 12. [latex]b\left(x\right)=\frac{1}{x+3}[/latex] on [latex]\left[1,1+h\right][/latex] 13. [latex]j\left(x\right)=3{x}^{3}[/latex] on [latex]\left[1,1+h\right][/latex] 14. [latex]r\left(t\right)=4{t}^{3}[/latex] on [latex]\left[2,2+h\right][/latex] 15. [latex]\frac{f\left(x+h\right)-f\left(x\right)}{h}[/latex] given [latex]f\left(x\right)=2{x}^{2}-3x[/latex] on [latex]\left[x,x+h\right][/latex] For exercises 16–17, consider the graph of [latex]f[/latex]. 16. Estimate the average rate of change from [latex]x=1[/latex] to [latex]x=4[/latex]. 17. Estimate the average rate of change from [latex]x=2[/latex] to [latex]x=5[/latex]. For exercises 18–21, use the graph of each function to estimate the intervals on which the function is increasing or decreasing. 18. 19. 20. 21. For exercises 22–23, consider the graph shown below. 22. Estimate the intervals where the function is increasing or decreasing. 23. Estimate the point(s) at which the graph of [latex]f[/latex] has a local maximum or a local minimum. For exercises 24–25, consider the graph below. 24. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing. 25. If the complete graph of the function is shown, estimate the absolute maximum and absolute minimum. 26. The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. What was the average rate of change of annual sales (a) between 2001 and 2002, and (b) between 2001 and 2004?

Year	Sales (millions of dollars)
1998	201
1999	219
2000	233
2001	243
2002	249
2003	251
2004	249
2005	243
2006	233

27. The table below gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population (a) between 2002 and 2004, and (b) between 2002 and 2006?

Year	Population (thousands)
2000	87
2001	84
2002	83
2003	80
2004	77
2005	76
2006	78
2007	81
2008	85

For the following exercises, find the average rate of change of each function on the interval specified. 28. [latex]f\left(x\right)={x}^{2}[/latex] on [latex]\left[1,\text{ }5\right][/latex] 29. [latex]h\left(x\right)=5 - 2{x}^{2}[/latex] on [latex]\left[-2,\text{4}\right][/latex] 30. [latex]q\left(x\right)={x}^{3}[/latex] on [latex]\left[-4,\text{2}\right][/latex] 31. [latex]g\left(x\right)=3{x}^{3}-1[/latex] on [latex]\left[-3,\text{3}\right][/latex] 32. [latex]y=\frac{1}{x}[/latex] on [latex]\left[1,\text{ 3}\right][/latex] 33. [latex]p\left(t\right)=\frac{\left({t}^{2}-4\right)\left(t+1\right)}{{t}^{2}+3}[/latex] on [latex]\left[-3,\text{1}\right][/latex] 34. [latex]k\left(t\right)=6{t}^{2}+\frac{4}{{t}^{3}}[/latex] on [latex]\left[-1,3\right][/latex] For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. 35. [latex]f\left(x\right)={x}^{4}-4{x}^{3}+5[/latex] 36. [latex]h\left(x\right)={x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}-1[/latex] 37. [latex]g\left(t\right)=t\sqrt{t+3}[/latex] 38. [latex]k\left(t\right)=3{t}^{\frac{2}{3}}-t[/latex] 39. [latex]m\left(x\right)={x}^{4}+2{x}^{3}-12{x}^{2}-10x+4[/latex] 40. [latex]n\left(x\right)={x}^{4}-8{x}^{3}+18{x}^{2}-6x+2[/latex] 41.The graph of the function [latex]f[/latex] is shown below. Based on the calculator screenshot, the point [latex]\left(1.333,\text{ }5.185\right)[/latex] is which of the following? A) a relative (local) maximum of the function B) the vertex of the function C) the absolute maximum of the function D) a zero of the function. 42. Let [latex]f\left(x\right)=\frac{1}{x}[/latex]. Find a number [latex]c[/latex] such that the average rate of change of the function [latex]f[/latex] on the interval [latex]\left(1,c\right)[/latex] is [latex]-\frac{1}{4}[/latex]. 43. Let [latex]f\left(x\right)=\frac{1}{x}[/latex] . Find the number [latex]b[/latex] such that the average rate of change of [latex]f[/latex] on the interval [latex]\left(2,b\right)[/latex] is [latex]-\frac{1}{10}[/latex]. 44. At the start of a trip, the odometer on a car read 21,395. At the end of the trip, 13.5 hours later, the odometer read 22,125. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip? 45. A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read exactly 3:40 p.m. At this time, he started pumping gas into the tank. At exactly 3:44, the tank was full and he noticed that he had pumped 10.7 gallons. What is the average rate of flow of the gasoline into the gas tank? 46. Near the surface of the moon, the distance that an object falls is a function of time. It is given by [latex]d\left(t\right)=2.6667{t}^{2}[/latex], where [latex]t[/latex] is in seconds and [latex]d\left(t\right)[/latex] is in feet. If an object is dropped from a certain height, find the average velocity of the object from [latex]t=1[/latex] to [latex]t=2[/latex]. 47. The graph below illustrates the decay of a radioactive substance over [latex]t[/latex] days. Use the graph to estimate the average decay rate from [latex]t=5[/latex] to [latex]t=15[/latex].