 So we've seen that the first derivative can be used to show where functions increase and are decreasing and where the extreme occur Well, this turns out the second derivative gives us the rate of change of the first derivative It indicates how fast the function is increasing or decreasing So the rate of change of the derivative affects the graph So think about that for a second the derivative the first derivative is the rate of change of the original function The second derivative is the rate of change of the rate of change So if we know how quickly the change is happening then that also affects the shape of the graph and this leads to a concept We've seen previously which we called Concavity a function is concave upward on the interval a to b if the graph of the function lies Above its tangent lines Let's think about that for a second If you had a graph that is concave up the way we usually think of it because as we talked about this in a Pre-calculus notion at the beginning of this lecture series We we understood concave up was to suggest that like your bucket is holding water Right here the curve shape is going up now It turns out that the proper definition of concavity is to do with tangent lines If your function is curving upward concave upward then the tangent line is Actually below the function Okay On the other hand if you are concave downward on the interval a to b then that occurs if the Tangent line is below the function Okay, and so the usual idea we have for concave down would look something like this So this is our concave upward and this is an example of our concave downward And so this is describing the curvature of the graph you could think of it If this was like a bucket of water and while the water would be coming down because it's not holding It's not holding the water whatsoever, but where are the tangent lines here? If I were to draw this function if I that is if I drew a tangent line of the function You see that the graph is concave down if the tangent line is above the function So let's use Desmos as a tool to experiment with this idea of concavity You can see on the screen this bluish green curve This is the function f of x equals x cubed minus 3x This is just meant as an example and you can also see this orange line on the screen This is the tangent line for our function and I've programmed it so that as I move the point of tangency It'll recalculate the the tangent line in real time So you can see me manipulating that the point and thus the tangent line moves along with it Okay, so we've explained with this example before notice as as we go from increasing to decreasing You can see the tangent line switch its sign from positive to negative there That meant you have a local maximum as it's decreasing then to increasing you hit a local minimum The derivative switch from positive to negative that we saw with the first derivative test But what about monotonicity now notice in this region right here where my tangent line is located the tangent line is above The function so that tells us that we are concave downward and in fact We're concave downward all the way here when we pass the local maximum our tangent line is still above the function We are still concave downward It doesn't seem to switch until about x equals zero and x equals zero It seems to switch sides notice now that my tangent line is a is below the curve And it will continue to be below the curve forever afterwards. So we see that When you are to the left of x equals zero the tangent line is above the curves That means we are concave downward when you pass x equals zero that means our my tangent line is actually now below the Curve and we are would be concave upward. There's something special about this point x equals zero This is what we call a point of inflection an inflection point is where you switch your concavity We switch from being concave down to concave up the inflection point is where the tangent line is it will switch sides It switches from being above to being below the curve like so And so I claim that this idea of concavity has something to do with the second derivative So notice what's happening to my tangent line if I start right here My tangent line has been it's really steep So that's going to be a large positive value as I get closer and closer and closer to This local maximum the tangent slope is getting smaller and smaller and smaller and smaller It's x then the tangent slope becomes zero. So the first derivative got smaller So even though the first derivative is positive in this interval the first derivative is decreasing I'm not saying the function f is decreasing. I'm saying f prime is decreasing Okay, once you get past the local maximum you now look at your slope there. It's a negative slope Um, and then what happens as we move along further further further as we go along It's going to get more negative more negative more negative until you hit x equals zero then x equals zero It's going to start to go up again, right? So when you're to the left of zero your your tangent line is falling falling falling That suggests that the second derivative is negative because if the tangent lines are falling That means my first derivative is decreasing Okay, once you get past zero notice the derivative the tangent lines are now rising They're going up and up and up and up that would suggest that the first derivative is increasing Now if the first derivative is increasing that means the second derivative is positive So let's summarize what we've seen Um in this with what we call the test of concavity here Let f be a function with derivatives f prime and f double prime Existing at all points in the interval a to b If then f is going to be concave upward on the interval exactly when the second derivative is positive Because an increasing derivative means that you're curving upward And um the function will be concave downward if the second derivative is negative And at an inflection point for a function f the second derivative It's going to be zero or does not exist So that is the inflection points the places where it switches concavitys those will occur at the critical numbers of the first derivative And so we're going to typically call those potential Uh the potential points Oh, excuse me. I can spell any of these words today potential points of inflection So the critical numbers of the first derivative we're going to call potential points of inflection aren't just ppis for short A ppi is a potential point of inflection The reason we say that is that critical numbers are not always extrema But every extremum is going to be a critical number The that same is also true for these points of inflection The points of inflection are going to be critical numbers of the first derivative That is what makes the second derivative equal to zero r d and e But not every critical number of the first derivative will be a point of inflection So we'll call them ppis as opposed to calling them critical numbers again The critical numbers are would make the first derivative go to zero r d and e the ppis are what makes the second derivative Uh b zero r does not exist Let's illustrate this with an example. Let's find all the intervals where f of x Which is equal to x to the fourth minus 8 x cube plus 18 x squared is concave upward or concave downward And let's find all of the the points of inflection So we have to calculate the second derivative to learn things about concavity So the first derivative by the power rule we're going to get 4 x cubed minus 24 x squared Plus 36 x We don't actually really need to worry about the critical numbers because we're not asking about monotonicity here So we're going to proceed to calculate the second derivative then Which by the same calculation technique as before we're going to get the second derivatives 12 x squared minus 48 x plus 36 This is a polynomial. It's never going to be undefined. So we do need to see when it's equal to zero We can factor out a coefficient factor of 12 Leaving behind x squared minus 4 x plus 3 So we need factors of 3 that have to be negative 4 So we can get x minus 3 and x plus x minus 1 excuse me Negative 3 times negative 1 is positive 3 but negative 3 plus negative 1 is equal to the negative 4 And so these are going to be our potential points of inflection our ppis This is equal to 3 and 1 So we're going to build a number chart a sign chart just like we did with we did first derivative tests to terminate monotonicity in extrema It's basically the same idea. We're just now using the second derivative So we're going to be concerned with the ppi 1 and the ppi 3. So let's look at the factors of the Second derivative. Well, one of the factors is 12. That's always going to be positive So it's not too consequential then there's the next factor of x minus 3 It's a it's an increasing linear function will be negative until it hits its x intercept Which is at 3 so it'll switch to be positive if you look at x minus 1 It'll be negative until it hits x equals 1 then it switches to be positive So then when we put all these factors together the second derivative You get positive times negative times negative. That's a positive You get positive times negative times positive. That's a negative and you're going to get a triple positive, which is a positive what this tells you about your first derivative is that The first derivative will be increasing When you're less than 1 it'll be decreasing when you're between 1 and 3 and it'll be Increasing when you're past 3 but we really don't care about the first derivative here What we care about is the function f itself If the second derivative is positive that means that our function function is concave upward If the second derivative is negative that means our function is concave downward and if our Second derivative is positive. It's going to be concave upward right there So what we see is the following so we see that f is concave upward It's concave upward on the interval We're going to get negative infinity to 1 union 3 to infinity We also see That it's going to be concave downward on the interval 1 to 3 So now we've identified the intervals where the function is concave upward and concave downward. What about points and inflection? Well, if you're going from concave up to concave down Right, if you're concave up to concave down, that's going to be a switch there. So we switch the signs So basically we get something like concave up to concave down that looks like an inflection But promise we don't actually know is the function decreasing or is it increasing right here? Because we actually didn't look into that information because it couldn't look something like this. Maybe it was like decreasing But maybe it's like concave up increasing What would something look like like that? It could be something doing something like this We don't actually know but the point is we can look at the first derivative to figure out that information But what we're going to do right now is just note if you switch from concave up to concave down That's an inflection if you switch from concave concave down to concave up. That's likewise an inflection here So we see that f has inflection points has inflection points At the values x equals 1 and x equals 3 So one critical thing we want to mention here is that the points of inflection of f are exactly the local extrema of the first derivative And these are going to be the places where the second derivative changes its signs