The Shape of a Graph, Part II

The Shape of a Graph, Part II

In the previous section we saw how we could use the first derivative of a function to get some information about the graph of a function. In this section we are going to look at the information that the second derivative of a function can give us a about the graph of a function.

Before we do this we will need a couple of definitions out of the way. The main concept that we�ll be discussing in this section is concavity. Concavity is easiest to see with a graph (we�ll give the mathematical definition in a bit).

So a function is concave up if it �opens� up and the function is concave down if it �opens� down. Notice as well that concavity has nothing to do with increasing or decreasing. A function can be concave up and either increasing or decreasing. Similarly, a function can be concave down and either increasing or decreasing.

It�s probably not the best way to define concavity by saying which way it �opens� since this is a somewhat nebulous definition. Here is the mathematical definition of concavity.

Definition 1

To show that the graphs above do in fact have concavity claimed above here is the graph again (blown up a little to make things clearer).

So, as you can see, in the two upper graphs all of the tangent lines sketched in are all below the graph of the function and these are concave up. In the lower two graphs all the tangent lines are above the graph of the function and these are concave down.

Again, notice that concavity and the increasing/decreasing aspect of the function is completely separate and do not have anything to do with the other. This is important to note because students often mix these two up and use information about one to get information about the other.

There�s one more definition that we need to get out of the way.

Definition 2

Now that we have all the concavity definitions out of the way we need to bring the second derivative into the mix. We did after all start off this section saying we were going to be using the second derivative to get information about the graph. The following fact relates the second derivative of a function to its concavity. The proof of this fact is in the Proofs From Derivative Applications section of the Extras chapter.

Fact

Notice that this fact tells us that a list of possible inflection points will be those points where the second derivative is zero or doesn�t exist. Be careful however to not make the assumption that just because the second derivative is zero or doesn�t exist that the point will be an inflection point. We will only know that it is an inflection point once we determine the concavity on both sides of it. It will only be an inflection point if the concavity is different on both sides of the point.