Learning Objectives
In this section, you will:
- Find intervals for concavity.
- Identify inflection points.
- Determine concavity’s relationship to average rates of change.
As part of exploring how functions change, it is interesting to explore the graphical behavior of functions. Concavity describes the shape of the function and how it is changing.
Consider the graphs below that show the total sales, in thousands of dollars, for two companies over 4 weeks.
As you can see, the sales for each company are increasing, but they are increasing in very different ways. Company A has a lot of sales immediately which could represent the release of a much anticipated product and then the increase in sales levels off. Company B starts with slower sales perhaps representing an unknown product which then starts to sell more rapidly perhaps because of word of mouth. To describe the difference in behavior, we can investigate how the average rate of change varies over different intervals. Using tables of values, we can find the average rate of change between consecutive points. For example, in Company A, we can use the first pair of points to get the average rate of change [latex]\frac{5-0}{1-0}=5[/latex] and the second pair of points to get the average rate of change [latex]\frac{7.1-5}{2-1}=2.1.[/latex]
From the tables, we can see that the rate of change for company A is decreasing, while the rate of change for company B is increasing.
For an increasing function, when the rate of change is decreasing, as with Company A, we say the function is concave down. For an increasing function, when the rate of change is increasing, as with Company B, we say the function is concave up.
Definition
A function is concave up if the rate of change is increasing.
A function is concave down if the rate of change is decreasing.
A point where a function changes from concave up to concave down or vice versa is called an inflection point.
Example 1: Describe the Concavity
An object is thrown from the top of a building. The object’s height in feet above ground after t seconds is given by the function [latex]h(t)=144-16{{t}^{2}}[/latex] for [latex]0\le t\le 3[/latex]. Describe the concavity of the graph.
Example 2: Concavity from a Table of Values
The value, V, of a car after t years is given in the table below. Is the value increasing or decreasing? Is the function concave up or concave down?
| t | 0 | 2 | 4 | 6 | 8 |
| V(t) | 28000 | 24342 | 21162 | 18397 | 15994 |
Try It #1
Is the function described in the table below concave up or concave down?
| x | 0 | 5 | 10 | 15 | 20 |
| g(x) | 10000 | 9000 | 7000 | 4000 | 0 |
Graphically, concave down functions bend downwards like a frown, and concave up function bend upwards like a smile.
Example 3: Determine Intervals of Concavity from a Graph
From the graph shown, estimate the intervals on which the function is concave down and concave up.
Try It #2
Create a graph of $$f(x)={{x}^{3}}-6{{x}^{2}}-15x+20$$ and use it to estimate the intervals on which the function is concave up and concave down.
Behaviors of the Toolkit Functions
We will now return to our toolkit functions and discuss their graphical behavior.
| Function | Increasing/Decreasing | Concavity |
| Constant Function
[latex]f(x)=c[/latex] |
Neither increasing nor decreasing | Neither concave up nor down |
| Identity Function
[latex]f(x)=x[/latex] |
Increasing | Neither concave up nor down |
| Quadratic Function
[latex]f(x)={{x}^{2}}[/latex] |
Increasing on [latex](0,\infty )[/latex]
Decreasing on [latex](-\infty ,0)[/latex] Minimum at x = 0 |
Concave up
[latex](-\infty ,\infty )[/latex] |
| Cubic Function
[latex]f(x)={{x}^{3}}[/latex] |
Increasing | Concave down on [latex](-\infty ,0)[/latex]
Concave up on [latex](0,\infty )[/latex] Inflection point at (0,0) |
| Reciprocal
[latex]f(x)=\frac{1}{x}[/latex] |
Decreasing [latex](-\infty ,0)\cup (0,\infty )[/latex] | Concave down on [latex](-\infty ,0)[/latex]
Concave up on [latex](0,\infty )[/latex]
|
| Reciprocal squared
[latex]f(x)=\frac{1}{{{x}^{2}}}[/latex] |
Increasing on [latex](-\infty ,0)[/latex]
Decreasing on [latex](0,\infty )[/latex] |
Concave up on [latex](-\infty ,0)\cup (0,\infty )[/latex] |
| Cube Root
[latex]f(x)=\sqrt[3]{x}[/latex] |
Increasing | Concave down on [latex](0,\infty )[/latex]
Concave up on [latex](-\infty ,0)[/latex] Inflection point at (0,0) |
| Square Root
[latex]f(x)=\sqrt[{}]{x}[/latex] |
Increasing on [latex](0,\infty )[/latex] | Concave down on [latex](0,\infty )[/latex] |
| Absolute Value
[latex]f(x)=\left| x \right|[/latex] |
Increasing on [latex](0,\infty )[/latex]
Decreasing on [latex](-\infty ,0)[/latex] |
Neither concave up or down
|
Key Concepts
- Concavity describes the shape of the curve. If the average rates are increasing on an interval then the function is concave up and if the average rates are decreasing on an interval then the function is concave down on the interval.
- A function has an inflection point when it switches from concave down to concave up or visa versa.
- Given a graph, intervals of concavity can be estimated by determining where the graph bends up versus where it bends down.
- Input values are used when describing intervals of concavity. Endpoint in the interval are not included so the notation uses parenthesis not square brackets.
