Concavity and Inflection Points

Earlier I introduced the second derivative test to help determine whether a critical point is a maximum, minimum, or neither. What we saw was that functions with a negative second derivative have the concave down shape like the classic example y = -x².

Functions with a positive second derivative have the concave up shape like the classic example y = x².

Points where the second derivative is 0 mark transitions between regions of differing concavity. These points are inflection points. The classic example of a function with an inflection point is y = x³ + 1 at x = 0.

An Outline of the Procedure for Graphing a Function

Steps you should go through to construct the graph of a function.

1. Check for vertical asymptotes and/or regions where the function is not defined. Use one-sided limits to determine the behavior of the function near any vertical asymptotes.
2. Determine the behavior at . Are there any horizontal asymptotes?
3. Compute the first derivative and use it to determine where the critical points are and where the function is increasing or decreasing.
4. Compute the value of the function at each critical point.
5. Compute the second derivative and use it to find inflection points and determine where the function is concave up or concave down.
6. Put together all of this information and sketch the graph.

Examples

Example One

Graph the function

1) The function is defined everywhere and has no vertical asymptotes.

2) Since the leading term is -x³ and the behavior of the leading term dominates the behavior of the function at infinity, we have

3) The first derivative is

To find critical points we do

and see that there are critical points at x = 1 and x = 5. It is easy to see that

4) f(1) = -5, f(5) = 27

5) The second derivative

is 0 at x = 3 and is negative for x > 3 and positive for x < 3.

6) Here is the graph.

Example Two

1) The function has a vertical asymptote at x = 1. To compute the behavior near that asymptote we do

2) To determine the limits at infinity we divide top and bottom by the leading terms.

3), 4) The derivative is

The derivative is never 0, so there are no critical points. The derivative is negative for all x, so the function is decreasing everywhere.

5) The second derivative is

The second derivative is negative when x < 1 and positive when x > 1.

6) The graph:

Example Three

1) The function is not defined for x > 5. There are no vertical asymptotes.

2) Since the square root is positive where it is defined, the behavior of the function at infinity is determined by the factor of x.

3) The first derivative is

There is a single critical point at x = 10/3, which falls in the domain of the function.

For x > 10/3 the first derivative is negative. For x < 10/3 the first derivative is positive.

4) The value of the function at the criticial point is

5) The second derivative is

There is an inflection point at x = 20/3, but the function is not defined there. For x < 5 the second derivative is negative, making the function concave down everywhere that it is defined.

6) The plot:

Homework

Section 4.3: 23, 24, 39, 40, 69, 70ab