Calculus: Use 2nd Derivative to Examine Concavity

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Curve Sketching
Concavity
Using the Second Derivative to Examine Concavity Page [1 of 4]
Okay. Now we just took a look at the notion of curvature, and in particular, the notion of a curve being concave up. And that notion requires us to notice that the function that produces the slopes when this is concave up, that function is increasing, which means that that function must have a positive derivative anywhere here. But that function is a derivative, and so, in fact, the second derivative--the derivative of the derivative--must be positive whenever the curve is concave up.
Similarly, for a curve to be curved concave down, the derivative function must be decreasing. And so, therefore, the derivative of the derivative must be negative. So, the second derivative is negative for concave down. The second derivative must be positive for concave up. And for changes in concavity, we must have the concavity being zero or the concavity not being defined, just like with finding critical points.
Okay, well I thought that--this is--this, I think, really is a tricky notion, by the way. And to be completely honest, I think this really requires a little bit of time for you, by yourself, to chew on this. If you re-think or re-watch the lecture and work it through yourself and realize this idea of taking the derivative of the derivative and realizing what that means. The first derivative represents slopes of tangents, but then, how that function is changing--instantaneous rate of change--is the derivative of that, and that gives you the second derivative.
Let's just jump to some examples so you can start seeing these notions in action. I want to--in the spirit that we've been keeping here--consider the examples in order that we've already looked at. The first one was this four x squared plus two x minus two. Let me see if I can very quickly recap for you what we've already seen.
We drew a little first derivative--a chart here, and we saw that there's a critical point at minus a fourth. We also saw that the derivative there was zero. To the left, it was negative, and to the right, it was positive. And so, we discovered a few things by the first derivative test--that's the fancy name for this. We see that here the function is falling, here the function is rising, and this, therefore, is a min. That's what we've already seen.
Now, I'd like us to examine the concavity. How is this thing curving? Well, so, let me just remind you what the derivative equals. That's eight x plus two. Now, for us to study the concavity, we need the second derivative. So, we need to now take the derivative of this function. The derivative of that function is equal to eight. It's a constant--eight.
But you'll notice that constant number is positive. It's positive for all x. So, since the second derivative is always positive, that means that the slopes are always increasing. And so, therefore, we must be concave up. And so, therefore, I conclude that f of x is always concave up because the second derivative is always positive.
Let me show you, actually, what the graph of this function looks like. I'm putting all the information together. I haven't plotted any of the points, but at least you can see that over here at minus a fourth--which if this is zero and this is minus ten. So, minus a fourth really is right here. We have that minimum. Notice that the function is falling and then rising, but always it's curved up. It's concave up.
So, now we know for sure that a parabola is always concave up, because it's second derivative is going to be constant. And if the parabola is a happy-faced parabola--it's a positive constant in front--we know it's always going to be curved up. We finally resolved that issue, because we know it can't wiggle. Because if it wiggles a little bit out there, well, there'd be changes in concavity. And we would find those changes in concavity.
But here we see the second derivative is always positive, and that means that, in fact, the derivatives are always getting bigger and bigger and bigger and bigger and bigger and bigger and bigger. That means this is concave up. Finally you know that for a fact.
Okay, let's try another example. Let's try the second example we looked at. Let me refresh your memory here. So, this was g of x equals x cubed minus three x squared minus nine x plus one. And we drew a little sign chart here. This is a little bit more elaborate because we had the point minus one and the point three. Those were the two places where the derivative equals zero...of x.
And it turned out, if you remember, how we saw that, in fact, here the derivative was positive. In this region, the derivative is negative, and this region, the derivative is positive. And so, we conclude, therefore, that the function is rising here, falling here, rising here, which means this is a max and this is located at a min. And that's what we concluded so far. That is also the first derivative test, basically--the fact that we're increasing then decreasing. We're using the first derivative to discover that's a max--decreasing, increasing. The derivative shows us this is a min.
I now want to look at the curvature. So, to look at the curvature, I need the second derivative. Let me remind you what the first derivative is. The first derivative of this is three x squared minus six x minus nine. But now, I need the derivative of the derivative in order to study the curvature. So, if I take the derivative of the derivative--the second derivative--I see that equals six x minus six.
Okay, well now to see when this thing is positive or negative, what I'm going to do is actually create an entire new sign chart just for the second derivative. So, the first thing I'm going to do is to find the place where this equals zero. So, I set this equal to zero to find the point of inflection. So, point of inflection, that's a candidate for where the curvature may change.
I set the second derivative equal to zero. When I do that, I see six x minus six equals zero. And if you solve that, you see that x equals one. So, x equals one is a candidate for being a place where the concavity might, in fact, change.
So, now I'm going to do something that looks a little deja vous-ish. I'm going to draw a sign chart, but not for the derivative, instead for the second derivative. I'm going to mark the points of inflection. There's only one, so I'm going to mark this one one. The second derivative there is equal to zero. This is not going to tell me if the function is increasing or decreasing. This is the second derivative. So, this will tell me where the curve is concave up--the second derivative positive--or concave down--the second derivative negative.
So, let's take a look on either side of this. If I pick a point to the left of one--let's say zero. If I plug in zero here, I see I get negative six. So, g double prime at zero equals negative six, but all I care about is the sign. It's negative. So, I mark this whole region as negative-land.
Could there be any other change there? No, because if there was another change--if it were positive-land there--there would have to be a place where either the second derivative doesn't exist or where the second derivative is zero, and I would have found that. So, this is the only candidate. So therefore, it must be all negative here.
What about to the right? Well, you could pick a point like two. The second derivative at two would equal six times two, which is twelve, minus six is six. That's positive. So, I see positive here. So, what does it mean for the second derivative to be negative and then the second derivative to be positive? That's not a fact about how the function is increasing or decreasing. This is a fact about the curvature of the function.
And we see that this is curved down. This is concave down here. Here's how I denote that. I draw these little concave down, sort of, sad faces to tell me that we're curving downward somehow. And here, I write concave up faces--happy faces--to tell me here we're concave up.
So, we're concave up here, and we're concave down in this region. Well, what does the graph of this look like? Well, let me show you. It actually looks like this. Now, notice that everything has been fulfilled here.
First of all, let's take a look at the increasing and decreasing issues. Notice that we do have a maximum at negative one, just like we predicted. And the function is increasing up to that maximum, and then decreasing all the way down to three, and then increasing from then on in. But also, notice how here we see this is all concave down in this region up to the point one. And notice that that point right there is a point where the concavity changes.
To the left, we see it's concave down, and to the right, we see it's concave up. So, it's concave up from this point onward, and concave down before that point, which is what is captured in this sign chart. Do you see that? So, here we see concave down up to one, and then we see concave up. So, that is actually an accurate picture of this cubic--of this cubic expression.
Okay, and now lastly, what I'd like to take a look at is the last problem in this little sequence of three that we've been following through our journey to this discovery. That's h of x equals x to the one-third power. Let me show you what the sign chart looked like for this. H prime of x. We saw we had one critical point. That was at zero, and that was a critical point because the derivative didn't exist. So, this is undefined here.
And we discovered that the derivative is positive to the right and to the left. So, we discovered that, in fact, this curve is increasing before and increasing after. So, this is neither a max nor a min. So, this is neither max nor min, neither a max nor a min.
Okay. Well, let's see about curvature. So, I need the second derivative. And so, first I need the first derivative. Let me remind you what the first derivative is. We already computed that. It's one over three x to the two-thirds. That's the way I've been liking to write it. But since I have to take the derivative of it, it might be more convenient to re-write it in the way we originally had it, which was with that negative exponent. That will make it easier for me to take the derivative.
Now, if I take the second derivative, what do I see? Well, I bring this down in front, which means I have a negative two-thirds times this one-third, which gives me a negative two-ninths x raised to what power? I take negative two-thirds, and I subtract one. So, I'm subtracting three over three, which gives me a minus five-thirds. And you could write this now as minus two divided by nine x to the five-thirds power. That negative sign pushes all this stuff in the denominator.
When is this fraction equal to zero? The only way a fraction can equal zero is if the numerator is zero, but the numerator is never zero. Negative two is not equal to zero, so this is never equal to zero.
Is there any place where this is undefined? Yes, this is undefined at zero. Because if I put a zero in here, I have a zero in the denominator. So, the second derivative is undefined at x equals zero. So, there's a candidate for maybe a point of inflection.
Let's take a look at that by looking at a sign chart for the second derivative. I'll plug in zero again here, and let's see what the second derivative is doing. Well, over here I need to pick a point. How about--oh, oh, wait. I think--I'm sorry, this g should be an h. The function's name is h. Little typo there.
So, h double prime--let's pick a point to the left. Let's pick, let's say, minus one. That's left of zero. What do I get when I plug that in here? Well, if I take minus one and take the cubed root, we get minus one. Now what's minus one to the fifth power. That's minus one times mines one times minus one times minus one times minus one. Since there's an odd number of minus ones I'm multiplying together that will give me a minus one times the nine gives me a minus nine. Minus two divided by minus nine is actually equal to two-ninths.
I'm not interested in the actual value. All I care about is if it's positive or negative. It's positive. So I put positive signs all over here. At this point, I put down as undefined. Now, what happens over here? Well, let's take a look at h double prime at a point to the right of zero. Let's say one.
If you plug in one, well, one to the five-thirds power is one. So, this equals negative two over nine. But all I care about, really, is the sign. It's negative. So, what does this mean? Any information about increasing or decreasing? No, the second derivative tells me about the curvature.
So, here I see this is concave up. So, these are all happy faces--concave up. Here is I see we have sad faces, so this is concave down somehow. So, it's concave up here, concave down here. So, this really is a point of inflection. This is a point of inflection, because the curvature changes.
Now, what does the graph of this look like? Well, to sketch the graph of this, we have to be a little careful, because the derivative is undefined here. So, we know that we're going to have something that really is weird, and it turns out the graph looks like this. Let's make sure that everything meshes.
Notice that, first of all, the function is always increasing. It's always going up. However, at this point, what's the tangent? Well, the tangent is actually completely vertical. The tangent is undefined. And what about concavity? Notice in this region, it's curving up. It's concave up. And in this region, it's curving down. It's concave down.
So, everything has been fulfilled. So you know at the point of inflection--concave up, then concave down. We're always increasing. And here, the derivative--the tangent--is undefined.
So, you could start to see how we can use critical points, where the function's increasing, decreasing, where the function has points of inflection, and the concavity to sketch a very accurate picture of this.
What I want to do next is a whole bunch of new examples where we do everything from start to finish, and you can see how we put together these really pretty pictures. I'll see you in the example section. Bye for now.