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This website has a video explaining how to do a second derivative test and has extra problems to practice.
-Ava Stanard

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- [Voiceover] So what I want to do in this video is familiarize ourselves with the second derivative test and before I even get into the nitty-gritty of it, I really just want to get an intuitive feel for what the second derivative test is telling us. So let me just draw some axes here. So let's say that's my y-axis, let's say this is my x-axis and let's say I have a function that has a relative maximum value at X equals C. So let's say we have a situation that looks something like that and X equals C is right over, so that's the point C, F of C. So if I can draw a straighter dotted line. So that is X being equal to C and we visually see that we have a local maximum point there and we can use our calculus tools to think about what's going on there. Well one thing that we know, we know that the slope of the tangent line, at least the way I've drawn it right over here, is equal to zero. So we could say F prime of C is equal to zero and the other thing we can see is that we are concave downward in the neighborhood around X equals C. So notice our slope is constantly decreasing and since your slope, notice it's positive, it's less positive, even less positive, it goes to zero, then it becomes negative, more negative and even more negative. So we know that F prime prime, we know that F prime prime of C is less than zero and so I haven't done any deep mathematical proof here, but if I have a critical point at X equals C, so F prime of C is equal to zero, and we also see that the second derivative there is less than zero. Well intuitively this makes sense that we are at a maximum value and we could go the other way if we are at a local minimum point at X equals C or relative minimum point. So our first derivative should still be equal to zero 'cause our slope of a tangent line right over there is still zero. So F prime of C is equal to zero. But in this second situation, we are concave upwards. The slope is constantly increasing. We have an upward opening bowl and so here we have a relative minimum value or we could say our second derivative is greater than zero. Visually we see it's a relative minimum value and we can tell just looking at our derivatives, at least the way I've drawn it, first derivative is equal to zero and we are concave upwards. Second derivative is greater than zero. And so this intuition that we hopefully just built up is what the second derivative test tells us. So it says hey look, if we're dealing with some function F, let's say it's a twice differentiable function. So that means that over some interval. So that means that you could find its first and second derivatives are defined and so let's say there's some point, X equals C, where its first derivative is equal to zero, so the slope of the tangent line is equal to zero, and the derivative exists in a neighborhood around C and most of the functions we deal with, if it's differentiable at C, it tends to be differentiable in the neighborhood around C and then we also assume that second derivative exists is twice differentiable. Well then we might be dealing with a maximum point, we might be dealing with a minimum point, or we might not know what we're dealing with and it might be neither a minimum or a maximum point. But using the second derivative test, if we take the second derivative and if we see that the second derivative is indeed less than zero, then we have a relative maximum point. Then so this is a situation that we started with right up there. If our second derivative is greater than zero, then we are in this situation right here, we're concave upwards. Where the slope is zero, that's the bottom of the bowl. We have a relative minimum point and if our second derivative is zero, it's inconclusive. We don't know what is actually going on at that point. We can't make any strong statement. So with that out of the way, let's just do a quick example just to see if this has gelled. Let's say that I have some twice differentiable function H and let's say that I tell you that H of eight is equal to five, I tell you that H prime of eight is equal to zero, and I tell you that the second derivative at X equals eight is equal to negative four. So given this, can you tell me whether the point eight comma five, so the point eight comma five, is it a relative minimum, relative minimum, maximum point or not enough info? Not enough info or inconclusive? And like always, pause the video and see if you can figure it out. Well we're assuming it's twice differentiable and I think it's safe to assume that and for the sake of our problem we're gonna assume that the derivative exists in a neighborhood around X equals eight. So in this example, C is eight. So point eight five is definitely on the curve. The derivative is equal to zero. So we're dealing potentially with one of these scenarios and our second derivative is less than zero. Second derivative is less than zero. So this threw us. So the fact that the second derivative, so H prime prime of eight is less than zero, tells us that we fall into this situation right over here. So just with the information they've given us, we can say that at the point eight comma five we have a relative maximum value or that this is a relative maximum point for this. If somehow they told us the second derivative was zero, then we would say it's inconclusive. If they told us and that's all they told us and if they told us the second derivative is greater than zero, then we would be dealing with a relative minimum value at X equals eight.

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Second derivative test (video) | Khan Academy

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Voiceover] So what I want to do in this video is familiarize ourselves with the second derivative test and before I even get into the nitty-gritty of it, I really just want to get an intuitive feel for what the second derivative test is telling us. So let me just draw some axes here.

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I found this resource very useful to read as it shows step by step processes of what you need to do to solve the problem. It uses all the notation that we are using in class and even has examples to go through to further deepen your understanding.
edit: -Emma McCabe

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Now that we have explained to death the concept of what concavity is, how do we test it? Well, we use the Second Derivative Test, which is almost exactly the same process as the First Derivative Test, with only one difference: we use the second derivative to test our function, rather than the first. The steps for the Second Derivative Test, then, are:

1. Find the second derivative of the function.

2. Find where the function is equal to zero, or where it is not continuous. Points of discontinuity show up here a bit more than in the First Derivative Test.

3. Define the intervals for the function.

4. Plug in a value that lies in each interval to the second derivative; if f '' (x) is positive, the function is concave upwards for that interval, and if f '' (x) is negative, the function is concave downwards for that interval.

As a note, any point at which the function changes concavity is called a point of inflection. Some textbooks and individual classes define them slightly differently, though: Some call any point that changes the concavity a point of inflection, and some do not include points of discontinuity, only points at which the function actually exists. Remember, a point of inflection is not necessarily all critical numbers, only ones which result in a change of concavity.
Now, let's try some of these:

Determine the regions in which the following function is concave upward or downward:

Now that we have a critical value, we know that our intervals are

Let's test the intervals to see if x = 2 is a point of inflection:

Since x = 2 changes the graph's concavity and is an actual point on f, it is a point of inflection.

Determine the regions in which the following function is concave upward or downward:

Now that we have the second derivative, we need to check for critical values. There is no x-value at which the second derivative can equal 0, however the function does not exist at x = 1, so that will be our only critical value. Let's test the intervals to either side of 1:

Even though the concavity changes at 2, it is not considered a point of inflection, since it is not an actual point on the graph, but rather a place where the function is undefined.

Determine the regions in which the following function is concave upward or downward. In this one, the numbers get pretty big, but don't let that intimidate you; we can solve it just like any other problem:

From here, we can see that the second derivative can't equal zero, but there are points of discontinuity at x = 3 and -3. It should also be noted, we don't necessarily have to factor and simplify a function all the way to the end; if you can see that part of a function won't equal zero, then it's perfectly fine to leave it as is, since factoring and simplifying won't change that. Just be careful, and make sure you absolutely know it can't equal zero first! Now, let's test out our intervals:

After that last one, let's finish this with an easier one. Determine the regions in which the following function is concave upward or downward.

From here, we see that our only critical value is x = 2. Let's test the intervals:

As you can see, even though x = 2 was a critical value for this one, it is not a point of inflection; f (x) is concave upwards for both intervals.

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Now that we have explained to death the concept of what concavity is, how do we test it? Well, we use the Second Derivative Test, which is almost exactly the same process as the First Derivative Test, with only one difference: we use the second derivative to test our function, rather than the first.

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This video provides an example of how to find the interval where a function is increasing or decreasing, and concave up or concave down.  The relative extrema and points of inflection are also found. (Hope)

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Ex: Determine Increasing / Decreasing / Concavity by Analyzing the Graph of a Function

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This video explains how to set up and solve a second derivative test in an easy-to-understand and concise way. It also quickly reviews how to do a first derivative test (in case you've already forgotten 😁).
–Brooke Hoctor

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This website provides a walkthrough on how to find the point of inflection. It provides definitions of other terms such as the second derivative, and has examples showing how to determine the point of inflection.

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An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa)

So what is concave upward / downward ?

Concave upward is when the slope increases:

Concave downward is when the slope decreases:

Here are some more examples:

Learn more at Concave upward and Concave downward.

Finding where ...

So our task is to find where a curve goes from concave upward to concave downward (or vice versa).

Calculus

Derivatives help us!

The derivative of a function gives the slope.

The second derivative tells us if the slope increases or decreases.

When the second derivative is positive, the function is concave upward.

When the second derivative is negative, the function is concave downward.

And the inflection point is where it goes from concave upward to concave downward (or vice versa).

Let's work out the second derivative:

And 30x + 4 is negative up to x = −4/30 = −2/15, positive from there onwards. So:

And the inflection point is at x = −2/15

A Quick Refresher on Derivatives

In the previous example we took this:

and came up with this derivative:

There are rules you can follow to find derivatives, and we used the "Power Rule":

Another example for you:

The derivative is: y' = 3x ² − 12x + 12

The second derivative is: y'' = 6x − 12

And 6x − 12 is negative up to x = 2, positive from there onwards. So:

And the inflection point is at x = 2:

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An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa) So what is concave upward / downward ? Concave upward is when the slope increases: Concave downward is when the slope decreases: Here are some more examples: Learn more at Concave upward and Concave downward.

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This video provides an in-depth summary and explanation of concavity, inflection points, first derivative and second derivative. The video also does a nice job of explaining the connections between these concepts.  You can work alongside the video to solve the problems with explanations as you go.
- Riley P

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Concavity, Inflection Points, Increasing Decreasing, First & Second Derivative - Calculus

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This website gives definitions for concavity, inflection points and the second derivative test. Along with notes that are simple to read, there are examples that require you to fill in the blanks to test your understanding. The website checks your answers as you go.
-Annabella Black

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Here we examine what the second derivative tells us about the geometry of functions.

Concavity

We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative tells us whether is increasing or decreasing at . We summarize the consequences of this seemingly simple idea in the table below:

If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. It is worth summarizing what we have seen already in a single theorem.

Test for Concavity Suppose that exists on an interval.

(a)

on that interval whenever is concave up on that interval.

(b)

on that interval whenever is concave down on that interval.

Let be a continuous function and suppose that:

Sketch a possible graph of .

Inflection points

If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up.

If the concavity of changes either from up to down or down to up at and has a tangent line at , then is an inflection point of

It is instructive to see some examples of inflection points:

It is also instructive to see some non-examples of inflection points:

We identify inflection points by first finding such that is zero or undefined and then checking to see whether does in fact go from positive to negative or negative to positive at these points.

Even if , the point determined by might not be an inflection point. You also need to check that concavity of changes on either side of

Describe the concavity of .

Note that we need to compute and analyze the second derivative to understand concavity, which can help us to identify whether critical points correspond to maxima or minima. Sometimes, rather than using the first derivative test for extrema, the second derivative test can also help you to identify extrema.

The second derivative test

Recall the first derivative test:

If to the left of and to the right of , then is a local maximum.

If to the left of and to the right of , then is a local minimum.

If changes from positive to negative, then is decreasing. In this case, might be negative, and if in fact is negative then is definitely decreasing, so there is a local maximum at the point in question. On the other hand, if changes from negative to positive, then is increasing. Again, this means that might be positive, and if in fact is positive then is definitely increasing, so there is a local minimum at the point in question. We summarize this as the second derivative test.

Second Derivative Test Suppose that is continuous on an open interval containing and that for some value of in that interval.

If , then has a local maximum at .

If , then has a local minimum at .

If , then the test is inconclusive. In this case, may or may not have a local extremum at , and the first derivative test should be used to find out.

In certain situations, when the second derivative is easy to calculate, the second derivative test is often the easiest way to identify local extrema. However, if the second derivative is difficult to calculate, you may want to stick with the first derivative test. Also, if has a critical point, at which is undefined, the second derivative test does not apply, which means you'll want to use the first derivative test instead. To add to this, even if the second derivative is easy to calculate, if it turns out that , then is neither concave up nor concave down at , so no conclusions can be made using concavity/the second derivative about whether corresponds to a local maximum or minimum.

Once again, consider the function Use the second derivative test, to locate the local extrema of .

If , what does the second derivative test tell us?

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Here we examine what the second derivative tells us about the geometry of functions. We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative tells us whether is increasing or decreasing at .

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https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6b/a/concavity-review

This website gives a multitude of practice problems and videos to help review the different parts of concavity.
- Ella Schaltz

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This video does a really good job of explaining concavity and how to find it in a short and concise way.

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How to Find Concavity in Calculus : Calculus Explained

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This video helps to clearly explain the upward or downward concavity. It also uses specific values in tables to display the slopes and uses several examples throughout the video. It has practice problems at the end as well.
- Jolie Brausch

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Concavity and the Second Derivative video

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Concavity and the Second Derivative video

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The video justifies the second derivative test as the way to determine relative minima and maxima. It also has a really helpful application example.
- Anavictoria Proaño

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- [Voiceover] So what I want to do in this video is familiarize ourselves with the second derivative test and before I even get into the nitty-gritty of it, I really just want to get an intuitive feel for what the second derivative test is telling us. So let me just draw some axes here. So let's say that's my y-axis, let's say this is my x-axis and let's say I have a function that has a relative maximum value at X equals C. So let's say we have a situation that looks something like that and X equals C is right over, so that's the point C, F of C. So if I can draw a straighter dotted line. So that is X being equal to C and we visually see that we have a local maximum point there and we can use our calculus tools to think about what's going on there. Well one thing that we know, we know that the slope of the tangent line, at least the way I've drawn it right over here, is equal to zero. So we could say F prime of C is equal to zero and the other thing we can see is that we are concave downward in the neighborhood around X equals C. So notice our slope is constantly decreasing and since your slope, notice it's positive, it's less positive, even less positive, it goes to zero, then it becomes negative, more negative and even more negative. So we know that F prime prime, we know that F prime prime of C is less than zero and so I haven't done any deep mathematical proof here, but if I have a critical point at X equals C, so F prime of C is equal to zero, and we also see that the second derivative there is less than zero. Well intuitively this makes sense that we are at a maximum value and we could go the other way if we are at a local minimum point at X equals C or relative minimum point. So our first derivative should still be equal to zero 'cause our slope of a tangent line right over there is still zero. So F prime of C is equal to zero. But in this second situation, we are concave upwards. The slope is constantly increasing. We have an upward opening bowl and so here we have a relative minimum value or we could say our second derivative is greater than zero. Visually we see it's a relative minimum value and we can tell just looking at our derivatives, at least the way I've drawn it, first derivative is equal to zero and we are concave upwards. Second derivative is greater than zero. And so this intuition that we hopefully just built up is what the second derivative test tells us. So it says hey look, if we're dealing with some function F, let's say it's a twice differentiable function. So that means that over some interval. So that means that you could find its first and second derivatives are defined and so let's say there's some point, X equals C, where its first derivative is equal to zero, so the slope of the tangent line is equal to zero, and the derivative exists in a neighborhood around C and most of the functions we deal with, if it's differentiable at C, it tends to be differentiable in the neighborhood around C and then we also assume that second derivative exists is twice differentiable. Well then we might be dealing with a maximum point, we might be dealing with a minimum point, or we might not know what we're dealing with and it might be neither a minimum or a maximum point. But using the second derivative test, if we take the second derivative and if we see that the second derivative is indeed less than zero, then we have a relative maximum point. Then so this is a situation that we started with right up there. If our second derivative is greater than zero, then we are in this situation right here, we're concave upwards. Where the slope is zero, that's the bottom of the bowl. We have a relative minimum point and if our second derivative is zero, it's inconclusive. We don't know what is actually going on at that point. We can't make any strong statement. So with that out of the way, let's just do a quick example just to see if this has gelled. Let's say that I have some twice differentiable function H and let's say that I tell you that H of eight is equal to five, I tell you that H prime of eight is equal to zero, and I tell you that the second derivative at X equals eight is equal to negative four. So given this, can you tell me whether the point eight comma five, so the point eight comma five, is it a relative minimum, relative minimum, maximum point or not enough info? Not enough info or inconclusive? And like always, pause the video and see if you can figure it out. Well we're assuming it's twice differentiable and I think it's safe to assume that and for the sake of our problem we're gonna assume that the derivative exists in a neighborhood around X equals eight. So in this example, C is eight. So point eight five is definitely on the curve. The derivative is equal to zero. So we're dealing potentially with one of these scenarios and our second derivative is less than zero. Second derivative is less than zero. So this threw us. So the fact that the second derivative, so H prime prime of eight is less than zero, tells us that we fall into this situation right over here. So just with the information they've given us, we can say that at the point eight comma five we have a relative maximum value or that this is a relative maximum point for this. If somehow they told us the second derivative was zero, then we would say it's inconclusive. If they told us and that's all they told us and if they told us the second derivative is greater than zero, then we would be dealing with a relative minimum value at X equals eight.

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Second derivative test (video) | Khan Academy

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Voiceover] So what I want to do in this video is familiarize ourselves with the second derivative test and before I even get into the nitty-gritty of it, I really just want to get an intuitive feel for what the second derivative test is telling us. So let me just draw some axes here.

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concavity, inflection, and second derivative

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concavity, inflection, and second derivative

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this video does a great job of explaining concavity in a concise way, as well as diving into second derivative in a more broken down and understandable form
- Natalia D'Souza

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Concavity, Inflection Points, and Second Derivative

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This has an in depth explanation of the second derivative test with a video, and practice problem you can submit answers to and it will tell you if you got it right and explain the correct answer

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The second derivative test is a method for classifying stationary points. We could also say it is a method for determining their nature. At a stationary point we calculate the value of the second derivative and its sign will help us determine whether we are dealing with a maximum or a minimum point.…

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Create a post-it note like this that describes your resource.  Link or paste your resource.  

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Create a post-it note like this that describes your resource.  Link or paste your resource.  

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This can be a tool, app, picture, website, video, anything!

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This video talks about the basics of inflections points and how to use the second derivative test to help determine where they are.
- Kendal Baldwin

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If you were paying close attention in the last video, an interesting question might have popped up in your brain. We have talked about the intervals over which the function is concave downwards. And then we talked about the interval over which the function is concave upwards. But we see here that there's a point at which we transition from being concave downwards to concave upwards. Before that point, the slope was decreasing, and then the slope starts increasing. The slope was decreasing and then the slope started increasing. So that's one way to look at it. Right here in our function we go from being concave downwards to concave upwards. When you look at our derivative at that point, our derivative went from decreasing to increasing. And when you look at our second derivative at that point, it went from being negative to positive. So this must have some type of a special name you're probably thinking. And you'd be thinking correctly. This point at which we transition from being concave downwards to concave upwards, or the point at which our derivative has a extrema point, or the point at which our second derivative switches signs like this, we call it an inflection point. And the most typical way that people think about how could you test for an inflection point, it's a point, well, conceptually, it's where you go from being a downward opening u to an upward opening u. Or when you go from being concave downwards to concave upwards. But the easiest test is it's a point at which your second derivative switches signs. So in this case, we went from negative to positive. But we could have also switched from being positive to negative. So inflection point, your second derivative f prime prime of x switches signs. Goes from being positive to negative or negative to positive. Switches signs. So this is a case where we went from concave downwards to concave upwards. If we went from concave upwards to concave downwards, like that, then this inflection point up until that point, the slope was increasing. So the second derivative would to be positive. And then the slope is decreasing, so your second derivative would be negative. So here your second derivative is going from positive to negative. Here your second derivative is going from negative to positive. In either case, you are talking about an inflection point.

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Inflection points introduction (video) | Khan Academy

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Inflection points are points where the function changes concavity, i.e. from being "concave up" to being "concave down" or vice versa. They can be found by considering where the second derivative changes signs. In similar to critical points in the first derivative, inflection points will occur when the second derivative is either zero or undefined.

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Concavity and the Second Derivative Test

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Concavity and the Second Derivative Test

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(Kiersten Moning) This site does a great job of walking you through important definitions and details regarding concavity, points of inflection, the Second Derivative Test, and relative minima and maxima. It also includes several sample problems with step-by-step solutions.

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Our study of "nice" functions continues. The previous section showed how the first derivative of a function, \(f'\), can relay important information about \(f\). We now apply the same technique to \(f'\) itself, and learn what this tells us about \(f\). The key to studying \(f'\) is to consider its derivative, namely \(f''\), which is the second derivative of \(f\). When \(f''>0\), \(f'\) is increasing. When \(f''<0\), \(f'\) is decreasing. \(f'\) has relative maxima and minima where \(f''=0\) or is undefined. This section explores how knowing information about \(f''\) gives information about \(f\).

Concavity

We begin with a definition, then explore its meaning.

Note: We often state that "\(f\) is concave up" instead of "the graph of \(f\) is concave up" for simplicity.

The graph of a function \(f\) is concave up when \(f'\) is increasing. That means as one looks at a concave up graph from left to right, the slopes of the tangent lines will be increasing. Consider Figure \(\PageIndex{1}\), where a concave up graph is shown along with some tangent lines. Notice how the tangent line on the left is steep, downward, corresponding to a small value of \(f'\). On the right, the tangent line is steep, upward, corresponding to a large value of \(f'\).
Figure \(\PageIndex{1}\): A function \(f\) with a concave up graph. Notice how the slopes of the tangent lines, when looking from left to right, are increasing.
If a function is decreasing and concave up, then its rate of decrease is slowing; it is "leveling off." If the function is increasing and concave up, then the rate of increase is increasing. The function is increasing at a faster and faster rate.

Now consider a function which is concave down. We essentially repeat the above paragraphs with slight variation.

The graph of a function \(f\) is concave down when \(f'\) is decreasing. That means as one looks at a concave down graph from left to right, the slopes of the tangent lines will be decreasing. Consider Figure \(\PageIndex{2}\), where a concave down graph is shown along with some tangent lines. Notice how the tangent line on the left is steep, upward, corresponding to a large value of \(f'\). On the right, the tangent line is steep, downward, corresponding to a small value of \(f'\).
Figure \(\PageIndex{2}\): A function \(f\) with a concave down graph. Notice how the slopes of the tangent lines, when looking from left to right, are decreasing.
If a function is increasing and concave down, then its rate of increase is slowing; it is "leveling off." If the function is decreasing and concave down, then the rate of decrease is decreasing. The function is decreasing at a faster and faster rate.

Note: A mnemonic for remembering what concave up/down means is: "Concave up is like a cup; concave down is like a frown." It is admittedly terrible, but it works.

Our definition of concave up and concave down is given in terms of when the first derivative is increasing or decreasing. We can apply the results of the previous section and to find intervals on which a graph is concave up or down. That is, we recognize that \(f'\) is increasing when \(f''>0\), etc.
Figure \(\PageIndex{3}\): Demonstrating the 4 ways that concavity interacts with increasing/decreasing, along with the relationships with the first and second derivatives.
Note: Geometrically speaking, a function is concave up if its graph lies above its tangent lines. A function is concave down if its graph lies below its tangent lines.

If knowing where a graph is concave up/down is important, it makes sense that the places where the graph changes from one to the other is also important. This leads us to a definition.

Definition: Point of Inflection

Figure \(\PageIndex{4}\) shows a graph of a function with inflection points labeled.
Figure \(\PageIndex{4}\): A graph of a function with its inflection points marked. The intervals where concave up/down are also indicated.
If the concavity of \(f\) changes at a point \((c,f(c))\), then \(f'\) is changing from increasing to decreasing (or, decreasing to increasing) at \(x=c\). That means that the sign of \(f''\) is changing from positive to negative (or, negative to positive) at \(x=c\). This leads to the following theorem.

We have identified the concepts of concavity and points of inflection. It is now time to practice using these concepts; given a function, we should be able to find its points of inflection and identify intervals on which it is concave up or down. We do so in the following examples.
Solution Solution
Recall that relative maxima and minima of \(f\) are found at critical points of \(f\); that is, they are found when \(f'(x)=0\) or when \(f'\) is undefined. Likewise, the relative maxima and minima of \(f'\) are found when \(f''(x)=0\) or when \(f''\) is undefined; note that these are the inflection points of \(f\).

What does a "relative maximum of \(f'\)" mean? The derivative measures the rate of change of \(f\); maximizing \(f'\) means finding the where \(f\) is increasing the most -- where \(f\) has the steepest tangent line. A similar statement can be made for minimizing \(f'\); it corresponds to where \(f\) has the steepest negatively--sloped tangent line.

We utilize this concept in the next example.

Not every critical point corresponds to a relative extrema; \(f(x)=x^3\) has a critical point at \((0,0)\) but no relative maximum or minimum. Likewise, just because \(f''(x)=0\) we cannot conclude concavity changes at that point. We were careful before to use terminology " possible point of inflection'' since we needed to check to see if the concavity changed. The canonical example of \(f''(x)=0\) without concavity changing is \(f(x)=x^4\). At \(x=0\), \(f''(x)=0\) but \(f\) is always concave up, as shown in Figure \(\PageIndex{11}\).
Figure \(\PageIndex{11}\): A graph of \(f(x) = x^4\). Clearly \(f\) is always concave up, despite the fact that \(f''(x) = 0\) when \(x=0\). It this example, the possible point of inflection \((0,0)\) is not a point of inflection.
Note: A note about concavity:

The Second Derivative Test

The first derivative of a function gave us a test to find if a critical value corresponded to a relative maximum, minimum, or neither. The second derivative gives us another way to test if a critical point is a local maximum or minimum. The following theorem officially states something that is intuitive: if a critical value occurs in a region where a function \(f\) is concave up, then that critical value must correspond to a relative minimum of \(f\), etc. See Figure \(\PageIndex{12}\) for a visualization of this.
Figure \(\PageIndex{12}\): Demonstrating the fact that relative maxima occur when the graph is concave down and relatve minima occur when the graph is concave up.
The Second Derivative Test relates to the First Derivative Test in the following way. If \(f''(c)>0\), then the graph is concave up at a critical point \(c\) and \(f'\) itself is growing. Since \(f'(c)=0\) and \(f'\) is growing at \(c\), then it must go from negative to positive at \(c\). This means the function goes from decreasing to increasing, indicating a local minimum at \(c\).
Solution
We have been learning how the first and second derivatives of a function relate information about the graph of that function. We have found intervals of increasing and decreasing, intervals where the graph is concave up and down, along with the locations of relative extrema and inflection points. In Chapter 1 we saw how limits explained asymptotic behavior. In the next section we combine all of this information to produce accurate sketches of functions.

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3.4: Concavity and the Second Derivative

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Our study of "nice" functions continues. The previous section showed how the first derivative of a function, \(f'\), can relay important information about \(f\). We now apply the same technique to \(f'\) itself, and learn what this tells us about \(f\).

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Second Derivative Test to Find Local Maximums and Minimums 

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Second Derivative Test to Find Local Maximums and Minimums 

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This video is a great resource that explains the steps to finding local maximums and minimums, plus it works through a few example problems.
- Kate Skufca

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Concavity and the Second Derivative Test 

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Concavity and the Second Derivative Test 

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This video has step-by-step examples on how to discuss concavity with either upward or downward concavity. Then, it also shows how to write the restrictions of the curve.
- Lily Ogburn

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Second Derivative Test examples step-by-step 

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(Kenzie Sickinger) How to explanation of the second derivative test with an introduction that puts this test's meaning into a real-world scenario. 

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(Kenzie Sickinger) How to explanation of the second derivative test with an introduction that puts this test's meaning into a real-world scenario. 

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This website is very in depth and helps fill in some gaps that I had as I was finishing the notes from 3.4. It breaks down all the steps and helps you to fully understand the process.

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This video help to show how to find different aspects of a graph using the graph itself. For example, critical numbers, upward concavity, and downward concavity. There are multiple examples in this video.
-Mallory Hanna

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Introduction into concavity and concave upwards and downwards intervals

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Introduction into concavity and concave upwards and downwards intervals

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Introduces the idea of concavity and how to describe the graph when it is concave up or concave down and how it relates to the second derivative of a function.

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What I have here in yellow is the graph of y equals f of x. Then here in this mauve color I've graphed y is equal to the derivative of f is f prime of x. And then here in blue, I've graphed y is equal to the second derivative of our function. So this is the derivative of this, of the first derivative right over there. And we've already seen examples of how can we identify minimum and maximum points. Obviously if we have the graph in front of us it's not hard for a human brain to identify this as a local maximum point. The function might take on higher values later on. And to identify this as a local minimum point. The function might take on the lower values later on. But we saw, even if we don't have the graph in front of us, if we were able to take the derivative of the function we might-- or even if we're not able to take the derivative of the function-- we might be able to identify these points as minimum or maximum. The way that we did it, we said, OK, what are the critical points for this function? Well, critical points are where the function's derivative is either undefined or 0. This is the function's derivative. It is 0 here and here. So we would call those critical points. And I don't see any points at which the derivative is undefined just yet. So we would call here and here critical points. So these are candidate points at which our function might take on a minimum or a maximum value. And the way that we figured out whether it was a minimum or a maximum value is to look at the behavior of the derivative around that point. And over here we saw the derivative is positive as we approach that point. And then it becomes negative. It goes from being positive to negative as we cross that point. Which means that the function was increasing. If the derivative is positive, that means that the function was increasing as we approached that point, and then decreasing as we leave that point, which is a pretty good way to think about this being a maximum point. If we're increasing as we approach it and decreasing as we leave it, then this is definitely going to be a maximum point. Similarly, right over here we see that the derivative is negative as we approach the point, which means that the function is decreasing. And we see that the derivative is positive as we exit that point. We go from having a negative derivative to a positive derivative, which means the function goes from decreasing to increasing right around that point, which is a pretty good indication, or that is the indication, that this critical point is a point at which the function takes on a minimum value. What I want to do now is extend things by using the idea of concavity. And I know I'm mispronouncing it. Maybe it's concavity. But thinking about concavity, we could start to look at the second derivative rather than kind of seeing just this transition to think about whether this is a minimum or a maximum point. So let's think about what's happening in this first region, this part of the curve up here where it looks like a arc where it's opening downward, where it looks kind of like an A without the cross beam or an upside down U. And then we'll think about what's happening in this kind of upward opening U part of the curve. So over this first interval, right over here, if we start over here the slope is very-- actually let me do it in a-- actually I'll do it in that same color, because that's the same color I used for the actual derivative. The slope is very positive. Then it becomes less positive. Then it becomes even less positive. It eventually gets to 0. Then it keeps decreasing. Now it becomes slightly negative, slightly negative, then it becomes even more negative, then it becomes even more negative. And then it looks like it stops decreasing right around there. So the slope stops decreasing right around there. And you see that in the derivative. The slope is decreasing, decreasing, decreasing, decreasing until that point, and then it starts to increase. So this entire section right over here, the slope is decreasing. And you see it right over here when we take the derivative. The derivative right over here, over this entire interval is decreasing. And we also see that when we take the second derivative. If the derivative is decreasing, that means that the second derivative, the derivative of the derivative, is negative. And we see that that is indeed the case. Over this entire interval, the second derivative is indeed negative. Now what happens as we start to transition to this upward opening U part of the curve? Well, here the derivative is reasonably negative. It's reasonably negative right there. But then it's still negative, but it becomes less negative and less negative and less negative, less negative and less negative, and less negative. Then it becomes 0. It becomes 0 right over here. And then it becomes more and more and more positive. And you see that right over here. So over this entire interval, the slope or the derivative is increasing. So the slope is increasing. And you see this over here. Over here the slope is 0. The slope of the derivative is 0. The derivative itself isn't changing right at this moment. And then you see that the slope is increasing. And once again, we can visualize that on the second derivative, the derivative of the derivative. If the derivative is increasing, that means the derivative of that must be positive. And it is indeed the case that the derivative is positive. And we have a word for this downward opening U and this upward opening U. We call this concave downwards. Let me make this clear. Concave downwards. And we call this concave upwards. So let's review how we can identify concave downward intervals and concave upwards intervals. So if we're talking about concave downwards, we see several things. We see that the slope is decreasing. Which is another way of saying that f prime of x is decreasing. Which is another way of saying that the second derivative must be negative. If the first derivative is decreasing, the second derivative must be negative, which is another way of saying that the second derivative over that interval must be negative. So if you have a negative second derivative, then you are in a concave downward interval. Similarly-- I have trouble saying that word-- let's think about concave upwards, where you have an upward opening U. Concave upwards. In these intervals, the slope is increasing. We have a negative slope, less negative, less negative, 0, positive, more positive, more positive, even more positive. So slope is increasing. Which means that the derivative of the function is increasing. And you see that right over here. This derivative is increasing in value, which means that the second derivative over an interval where we are concave upwards must be greater than 0. If the second derivative is greater than 0, that means that the first derivative is increasing, which means that the slope is increasing. We are in a concave upward interval. Now given all of these definitions that we've just given for concave downwards and concave upwards, can we come up with another way of identifying whether a critical point is a minimum point or a maximum point? Well, if you have a maximum point, if you have a critical point where the function is concave downwards, then you're going to be at a maximum point. Concave downwards, let's just be clear here, means that it's opening down like this. And when we're talking about a critical point, if we're assuming it's concave downwards over here, we're assuming differentiability over this interval. And so the critical point is going to be one where the slope is 0. So it's going to be that point right over there. So if you're concave downwards and you have a point where f prime of, let's say, a is equal to 0, then we have a maximum point at a. And similarly, if we're concave upwards, that means that our function looks something like this. And if we found a point, obviously a critical point could also be where the function is not defined, but if we're assuming that our first derivative and second derivative is defined here, then the critical point is going to be one where the first derivative is going to be 0. So f prime of a is equal to 0. And if f prime of a is equal to 0 and if we're concave upwards in the interval around a, so if the second derivative is greater than 0, then it's pretty clear, you see here, that we are dealing with a minimum point at a.

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What I have here in yellow is the graph of y equals f of x. Then here in this mauve color I've graphed y is equal to the derivative of f is f prime of x. And then here in blue, I've graphed y is equal to the second derivative of our function.

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This video gives a walkthrough on how to determine the concavity using an algebraic method. I love Khan Academy's videos when I need extra help or a brief recap on a subject!
-Caroline Lamermayer

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Analyzing concavity (algebraic) | AP Calculus AB | Khan Academy