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Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph. In Exercises $27-32$ use the calculator maximum-minimum feature to check the local maximum and minimum points.$$y=4 x^{3}-3 x^{4}+6$$

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 5

Using Derivatives in Curve Sketching

Derivatives

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everyone. So today we're gonna be finding the critical points of this function and then using the first Sir votive test to see what kind of critical points they are if they're local, Maxima minima are neither and number in a check by wrapping. So let's get started. The way that we find the critical points of a function is we take the derivative right. So the derivative of this function is 12 breaks abuse of power rules. We do four times three x to the third, minus 12 x squared, and then that six is just a plus zero. We need to write that down. Awesome. And so the way that we use distributive to see what the critical point shark is, we see where this derivative is equal to zero. That's gonna tell us where there's a big change. Intergraph, where these critical points are so do that. We just have to solve this function for zero. So to do that, let's do a couple of things first. The first thing that we can do with this well, let's go ahead 74 0 All right, and now let's see how we can factor it. The first thing we can Dio is factor out a 12 and a next squared right, cause there's a 12 and an expert in both of these And that leaves an X and the minus one. Right? And so we set both of these equal to zero. Never hear. The only way that this could be equal to zero is if X is equal to zero. And here we just bring one to the other side. We get X is equal to one, so these are gonna be are two critical points. But how are we gonna tell what happens in between them? What is the big change that takes place of both of these? And that is what we use the first derivative test for. So for the first derivative test, we have our number line, and we're gonna make put the critical points on our number 10 and one, and then we're gonna test the points in between them. Right, So 91 is a good, easy test over here. We're gonna use 1/2 in the middle, and then we'll use to on this side, right, Because we know what's happening here. Right here. The derivative is equal to zero. But on either side of these things, we want to know what the derivative is doing right, because if it's negative are derivative over here and negative one is negative, that means that our function is going down. It's decreasing, right? And so we want to know what happens at each part of this. So let's try it. First, we have our derivative burning plug in that negative one we can flip back. This is our derivative. That's what we're gonna be plugging it into. Or we could use the factored out version here. And that is actually we're going to be wearing use this factored out. First, we're gonna find that it's a little bit easier to work with. Okay, so we have 12 and we plug in our negative one squared, they one minus one. And so this native one squared, that's going to give us a positive 12. Right? That skirt is gonna, um, cancel out the negative. So we have a 12 times negative one minus one thing, too. But we don't really care what exactly these values are. We care about their signs. We just need to know if this function is increasing or decreasing if the derivative is positive or negative. So we have a positive and a negative, and that gives us and that you have right five times negative, negative, Awesome. Which means that at this interval, a negative one, our function is decreasing. So let's try the next one. We're in plug in 1/2 and we get well when have squared 1/2 minus one. Awesome. And so here It doesn't really matter what this 12 times 1/2 squared is we just want new It's positive, right? So whatever it is 12 times 1/2 squared And over here, this was pretty easy to figure out. That's gonna be a negative 1/2 right? And so here we have again, we have a positive and a negative, and that's going to give us a negative. And so we know at 1/2 were still decreasing. And since we're still decreasing, we know this critical point is gonna be where we change con cavity, right? We're not changing direction. We're not going from decreasing to increasing to get a local minimum or a maximum, right. It's just gonna change the shape of the function, that con cavity. All right, you have one more point to plug in. I tried to so 12. I'm soo swearing tu minus one. That's going to give us well, times four and one and you'll notice both of these things are positive, which means we're going to get a positive, which means that this interval, our function is increasing. So you can see here at this one, we do have a change in the direction, so we're gonna have a local minima at that point. And so let's look just at our number, line and all the information we need is right here to find which points are maximum Emma and which one or ones there either. Okay. And so, like, kind of discussed before at this zero. We don't really have anything. We have neither. We have a change in khan cavity because we're not changing our direction. We're going down the whole time, so X equals zero. We have neither, but in X equals one, we have a local minima from an imam. And we know that because we see that our graph goes from decreasing to increasing so that at 11 is our point where it changes that direction and so everything around it is going to be greater than it. Right? So these were critical points on let's check using a graphing calculator autographing application, whether we got this right. So this is from a graphing application. And if we can look back, you remember that at X equals zero, we have need there. We have attending cone cavity and at X equals one. We said we had a local minima look so that X equals zero. We do, in fact, have neither. We have a change in khan cavity, right, Because we're going down this whole time. But we have this change in the shape of the graph, right? And then at retractor zero, then at one I can't see it, but this is gonna be our one. We have a local minima, right? Everything around it is greater than it. And it changes that direction right at that one awesome job

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