# Concavity and the Second Derivative Test

If is a function, the second derivative of y (or of f) is the derivative of the first derivative. Notation:

Thus,

Example. Find the second derivatives of the following functions.

(a) .

(b) .

(c) .

The first derivative gives information about whether a function increases or decreases. In fact:

• A differentiable function increases on intervals where its derivative is positive, and vice versa.
• A differentiable function decreases on intervals where its derivative is negative, and vice versa.

A function is concave up on an open interval if is positive on the interval. And a function is concave down on an open interval if is negative on the interval.

A point where the concavity goes from up to down or from down to up is called an inflection point.

What do these conditions mean geometrically?

Consider the curve below.

The tangent line A has negative slope, the tangent line B has zero slope, and the tangent line C has positive slope. Therefore, as you move from left to right, the slope of the tangent line increases.

But the slope of the tangent line is given by , and to say something increases means its derivative is positive. So the derivative of --- which is --- must be positive. By the definition, this means the curve is concave up.

Now consider the curve below.

The tangent line at A has positive slope, the tangent line at B has zero slope, and the tangent line at C has negative slope. As you move from left to right, the slope of the tangent line decreases.

The slope of the tangent line is given by , and to say something decreases means its derivative is negative. So the derivative of --- which is --- must be negative. By the definition, this means the curve is concave down.

The two pictures exemplify the geometric meanings of concave up and concave down.

Example. The graph of a function is pictured below.

Determine the intervals on which the function is concave up and the intervals on which it is concave down. Find the x-coordinates of any inflection points.

The graph is concave up on , , and . The graph is concave down on .

Note that concavity is a property of a graph on an open interval, so the endpoints aren't included.

There are inflection points at and at .

Example. Find the intervals on which is concave up and the intervals on which it is concave down. Find the x-coordinates of any inflection points.

I set up a sign chart for , just as I use a sign chart for to tell where a function increases and where it decreases. The break points for my concavity sign chart will be the x-values where and the x-values where is undefined.

In this case, for and , and there are no points where is undefined. The break points are at and .

I picked numbers in each interval and plugged the numbers into . If is positive, I put a "+" on the interval and draw a concave-up curve below the interval; if is negative, I put a "-" on the interval and draw a concave-down curve below the interval.

The function is concave up for and for . It is concave down for . and are inflection points.

Example. Find the intervals on which is concave up and the intervals on which it is concave down. Find the x-coordinates of any inflection points.

for ; is undefined for .

The function is concave up for and for . It is concave down for . and are inflection points.

Concavity provides way to tell whether a critical point is a max or a min --- well, sometimes. This method is called the Second Derivative Test.

Consider a critical point where , i.e. where the tangent line is horizontal. Here are two possibilities.

The point A is a local max; it occurs at a place where the curve is concave down, i.e. where .

The point B is a local min; it occurs at a place where the curve is concave up, i.e. where .

To summarize, if , then"

• If , then is a local max.
• If , then is a local min.
• If , the test fails. Try the First Derivative Test.

Example. Use the Second Derivative Test to classify the critical points of .

The critical points are and .

, so is a local min.

, so the test fails at .

Here's the graph:

In fact, is neither a max nor a min.

Example. It is not true that if (so c is a critical point) and (so the Second Derivative Test fails), then is neither a max nor a min. To say the test fails means that you can draw no conclusion, and you need to do more work. The point could still be a max or a min!

For example, consider . Then and , so and . Thus, is a critical point, and the Second Derivative Test fails. Nevertheless, is a local min, as you can verify by using the First Derivative Test.

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