Concavity of Functions

Introduction

Concavity is all about the rate at which the slope of a curve is increasing or decreasing. It has many important applications in mathematics, not the least of which is to help you decide which part of a hill to cycle up. Unless you are feeling particularly masochistic, you probably don't want to cycle up parts of a hill where the slope is increasing at an increasing rate!

There are two types of concavity that are particularly useful in calculus: concave up and concave down. Let's try and untangle what these terms mean by drawing some pictures.

The following pictures show some more curves that would be described as concave up or concave down:

Of course, at this point, someone is going to pipe up and ask what happens when the slope of a curve neither increases nor decreases. In this case, the function is a straight line. As we'll see later, straight lines satisfy the definitions of both concave up and concave down. How's that for muddying the waters? Let's explore more of the gory details about concavity before we get too worried about that.

Finding where a curve is concave up or down

You guessed it, it isn't enough to know what concave up or concave down curves look like! We need to be able to find where curves are concave up or down. A curve can have some parts that are concave up and other parts that are concave down, and it's useful to be able to work out which is which, even before you draw the graph. An understanding of concavity comes in really handy when you need to be able to solve problems like finding the maximum time you can spend playing games on your iPad during a boring class before your teacher finds out... There are also applications in economics and many other fields.

Coming up with a definition (a.k.a. as keeping the mathematicians happy)

The mathematicians aren't satisfied if we say that a curve is concave up if it is increasing at an increasing rate or concave down if it is increasing at a decreasing rate. In fact, you shouldn't be satisfied either: parts of concave up curves can decrease at decreasing rates, and parts of concave down curves can decrease at increasing rates.

So... we first have to ask what's special about curves that are concave up or down. The key idea is that we if we draw a line joining any two points of the curve, it won't cross from one side of the curve to the other. Clear as mud? Hopefully, the pictures below will make this idea a little easier to understand.

Well, we're on the road to making the mathematicians happier! The next step is to chuck a formula into the mix. Let's use another picture to help us figure out what's going on. We start by choosing any two values of \(a\) and \(b\) that lie in the interval we're interested in and draw a line from \(f(a)\) to \(f(b)\): Now you can make the \(x\)-values move between \(a\) and \(b\) using a value (often called a parameter) \(t\), which lies between \(0\) and \(1\): Let's look at what happens to \(x\) for different values of \(t\):

• When \(t = 0\), \(x = 0a + 1b = b\)
• When \(t = 1\), \(x = 1a + 0b = a\)
• When \(t\) moves between \(0\) and \(1\), \(x\) moves between \(a\) and \(b\).

Now, what about the function values corresponding to these \(x\)-values? How do they compare to the points on the line joining \(f(a)\) to \(f(b)\)?
When \(x = ta + (1-t)b\):

• \(y=f(x)\) is equal to \(y = f(ta + (1 - t)b)\)
• The point on the line corresponding to \(x\) is \(y = tf(a) + (1 - t)f(b)\)

If the function is concave up, then the line needs to stay above or on the curve (and not cross it). So this means:

and if the function is concave down, then the line needs to stay below or on the curve (and not cross it). So we change \(\leq\) to \(\geq\) to give:

These are the definitions that mathematicians actually use to decide whether something is concave up or concave down.

How do I remember which is which?

Some people say that the best way to remember the shape of a concave up curve is using a cup like this:

But I think it's easiest to remember that a function is concave up if its mood is up, so it has a smiley face like this:

and concave down if it's feeling a big down like the mouth on this sad face:

Using calculus to help out

Remember, we can use the first derivative to find the slope of a function. However, we want to find out when the slope is increasing or decreasing, so we either need to look at the formula for the slope (the first derivative) and decide, or we need to use the second derivative.

• If a function is concave up, then its second derivative is positive.
• If a function is concave down, then its second derivative is negative.
• To find out what happens when the second derivative is zero, read the article on points of inflection.

Example:

Example

Finally, what about straight lines?

Earlier we mentioned that the slope of straight lines neither increases nor decreases: it always stays the same.

This means that you can choose to say that a straight line is concave up or concave down. A straight line \(f(x) = mx + b\) satisfies the definitions of both concave up and concave because we always have

Example: