What is an inflection point?
An inflection point is one of those key landmarks on a graph of a function where an important change in behavior occurs. Not all functions have inflection points. We will examine how to spot an inflection point on a graph, and what the significance of this point is.
An inflection point is a point on the graph where the concavity changes, either from concave up to concave down, or from concave down to concave up. In the image shown, the origin is an inflection point.
So we ask the question, how do you find the inflection point given an equation for a function, and under what conditions are we guaranteed that an inflection point exists?
Hopefully we recall from section 1.10 that a graph is concave up when
f ''(x) > 0, and a graph is concave down when f ''(x) < 0. It would stand to reason then that the change in concavity occurs when f ''(x) = 0, and this is quite often (but not always the case.)
So, to find an inflection point the first step is to find the second derivative, set it equal to zero, and solve for x, as the first example shows.
Does this pattern always seem to hold (continue to be true)? Is the second derivative output value always negative when the parent curve is concave down, and always positive when the parent curve is concave up? In a word, yes. To determine the shape or concavity of a graph at a point, plug the x value into the second derivative, as in the function from example 1, . If the resulting number is negative, the graph is concave down (like a frown.) If the number is positive, the graph is concave up (like a cup). If the second derivative is zero, we probably have a change in concavity. For our given function, f '' (-1) = -22, a negative value, which tells us that the graph is concave down. Substituting x = 1 into the second derivative, we get f '' (x) = 2, a positive value, indicating that the graph is concave up. (If the second derivative equals zero, we probably have a change in concavity. This point is called an inflection point.)
What would the inflection point be for the function ?
We set and solve. At x = 5/6, we have an inflection point, the graph changes concavity.
Summary regarding Inflection Points.
Provided that the second derivative exists over some interval, we follow these steps:
Find Intervals over which a function is concave up or down, and find any inflection points, if they exist.
Consider the function .
The first and second derivatives are: .
When we set the second derivative equal to zero and solve for x, we find two inflection points, and .
Next, we can find the x intervals on which the function is concave up and concave down by testing the sign of the second derivative near the inflection points.
So, for example, we might calculate the following. Please be lazy! Choose values on with side of the critical point that are easy to work with. The size doesn't matter, only the sign.
- f ''(-1) = +56 > 0
(notice the + sign on the interval to the left of in the picture above).
- f ''(0) = - 4 < 0
(observe the minus sign above the interval from to ).
- f ''(1) = +56 > 0
(again, we see a + sign on the interval to the right of ).
The answer: Let's summarize of our results:
The graph of is
- concave up on the following intervals : and
- concave down on the following interval:
Recall from other discussions that the critical points are not included in the intervals.
A production function is an equation that expresses the relationship between the quantities of production factors (such as labor and capital) used and the amount of product obtained as a result.
It gives the amount of product that can be obtained from every combination of factors, assuming that the most efficient available methods of production are used.
The production function can measure the marginal productivity of a particular factor of production and determine the cheapest combination of production factors that can be used to produce a given output.
The graph shown illustrates a production function for the ever popular but totally fictional product, Widgets.
Suppose the production function is given by .
Find the inflection point. That is, find where the rate of decrease in production begins to increase (that is, the rate of decrease begins to slow down). Then interpret the meaning of the inflection point.
When 2000 laborers are employed, 4 million widgets are produced. This is the point of increasing returns, where the amount of decrease begins to change for the better (amount of decrease begins to lessen) - note that this trend continues, causing slopes to become positive.
We also note that the inflection point is where the graph of f is decreasing at its fastest rate, that is, where we could expect to see an extreme value value for f ' .
This is indicated by the red tangent line at (2, 4).
Closely related to the general description of graphs as concave up or concave down, we find that the terms concavity and convexity are used widely in economic theory.
- A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point.
- Similarly, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point. These concepts are illustrated in the following figure.
In other words, when we are in the realm of economics, instead of saying concave up, we can use the term convex, and instead of concave down, we simply say, concave.
Determine where the following function is concave or convex (or neither) and determine any possible inflection points.
For what values of "c" will g(x) have two, one, or no inflection values?
Solution: First find the second derivative and set it equal to zero.
We see that this results in a quadratic equation. We recall that a quadratic equation can have two, one or no solutions depending upon the value of the discriminant. (click on the link if you have forgotten what the discriminant is.)
We see that
So we see that there are only two possibilities here since the discriminant can never be negative.
Note that "c" has been used in two different contexts here, once as a constant in the equation, and again as a standard value of the standard quadratic function ax˛ + bx + c = 0.