Geometry of Functions IX: Concavity

Primary tabs

Lesson Summary: 
In this lesson we define the concept of concavity and show in pictures when a function is concave up or concave down.
Lesson Inputs: 
Geometry of Functions VIII: The Second Derivative Test
Lesson Outputs: 
Geometry of Functions XI: Inflection Points
Geometry of Functions X: Determining Concavity From Second Derivatives

SAMPLE LESSON

Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:



Lesson: 

The concept of concavity is best demonstrated in a picture:


Demonstrating Concavity: Function of Concave Up and Concave Down Intervals


In the picture above, the part of the graph that is concave up is like a "cup" and the part of the graph where the graph is concave down is like the cup turned over. Here is our formal definition:

Concavity
Let $f(x)$ be a function that is differentiable on an open interval $I$. We define concave up and concave down as:
  • $f(x)$ is concave up on $I$ if and only if its derivative, $f'(x)$, increases on the interval $I$.
  • $f(x)$ is concave down on $I$ if and only if its derivative, $f'(x)$, decreases on the interval $I$.

We can gain some more intuition around our formal definition of concavity by looking at a specific graph and getting real numbers. For example, we know that the derivative of $x^2$ is $2x$. By looking at <Sign in to see all the formulas> we get the sense (visually) that $x^2$ is concave up. Let's look at the value of the derivative at $-5$, $0$, and $2$. (Note that there is nothing special about these values and that we have arbitrarily chosen them.) We have given the values and the tangent lines at those points in the graph below. We see that as we go from left to right that <Sign in to see all the formulas> is getting bigger ($-10$ to $0$ to $4$). Since the derivative is increasing $x^2$ is concave up. A formal proof that $x^2$ is concave up can be found here.

Concave Up x^2 with Tangent Lines


<Sign in to see all the formulas> is a great example of a function having the same concavity everywhere. But a function's concavity can change in different domains of the function. A good example of this is <Sign in to see all the formulas>. It is concave down from <Sign in to see all the formulas> and concave up from <Sign in to see all the formulas>. A proof of this can be found here. See the graph of $x^3$ below.

Concavity of x^3