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University of North Texas

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Problem 21 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = (x - 3)\sqrt{x} $

Answer

see solution

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Video Transcript

able to sketch the ground. Why is equal to X minus three times square root of X. So in this truck where they give us this laundry list of steps we should follow. So let's just go ahead and follow moves one by one. So the first thing they tell us to do it, So look for our domain of our function. So we know that underneath the radical we need this to be larger than or equal to zero. So they'll just be X greater than it was there or is a road to infinity. And he should actually be including zero. Since we can plug zero, I'm going next that the bracket there will ever make. The next thing they tell us to do is to look for our intercepts. So and her sums, but not eccentricity, just intercepts intercepts. So let's go ahead and first find our ex intercepts. So that's going to be where the function is just equal to zero X minus three. Busy for tickets were with bex, so that tells us either zero is equal to X minus three or, uh, zero. Is it to the square root of X? So that says X is equal to three. X equals there. And both of these airlifting are coming mix. We confined our Why intercept? And actually, we already know what are y intercept IDs from here. So this tells us that's also our wine. So I actually don't need to go about solving for that now. First symmetry. Well, there really won't be any kind of period of periodic city within this, but we could possibly check to see if f of negative X works to see if this is an even or odd function. But we've run into the issue of Won't do X minus three negative, X minus three and the square root of negative X. Well, if I try to plug in like one or two, this part here is going to be undefined or our domain undefined for domain. So we really won't have any symmetry, early snow, symmetry about the origin or the why access. Now, the next thing they tell us to look for is acid trips. So we won't have any murder class in tips, but we can at least possibly check to see what our behavior is going to be. So since we don't go too negative infinity. We can at least go ahead and check what happens as excuse to positivity limit has exposed to many of FX. Well, X minus three is gonna go to infinity and the square root of X is also going to go to infinity. So we have that are in behavior here is just gonna go towards. And as X approaches, the next thing they want us to do is to determine where are changeable is increasing and decreasing. And I'm gonna love this. In the next step of binding are local maxes and mittens. So to do that when you know what? Why private sorts to another peek. So first, right, why? Why is going to be equal And I'm going to distribute the square root and doing that is going to give X to the three haps minus three X to the 1/2. And I'm just rewriting it with the power so I can apply Power will quickly. So that's gonna be wide front is equal to three halves X to be one how minus three over two times X to the negative one hat. And if you want, we can go ahead and combine those two get three X minus one over to square roots of X. And I want to do this because now I can go ahead and set the secret zero to figure out our possible critical points so that that plug into denominator will ever make This equals zero. So I just have x minus one is zero or X is equal to one. So this is our possible point of infection. Now, something else we would want to check his where are derivative is undefined for critical values, which would be X is equal to zero. But since that is our in point of our domain, we don't really need to go to check to see if they'll be a local mid or Max. But do keep in mind normally, we also need to check for where the function is undefined for critical values as well. What? So why crime? We found to be three x minus one over to swear with sex. And so But we need to do what's fair words increasing. Which would be where Why, prime is strictly larger than here. So this is going to be from 1 to 10 and we know the function is going to be decreasing when why crime is strictly less than which is 0 to 1. Now we found that are possible point that would be a maximum is ex eagle one. So to the left of exit of the one, the function is decreasing and to the right of ecstasy, the one the function should be increasing. So that tells us this year is going to be a local minimum over function. The next thing they suggest that we do is to look for con cavity and any kind of points of inflection we might have. So that means when you look for wide double time, so why double Pond is going to be equal to now? I'm going to go ahead and take the derivative of this because it's just powerful, as opposed to when I used algebra to simplify. So doing that is going to give three worse X to the negative 1/2 and then it's going to be a plus re over or and then X to the negative, the rehabs, and now we could use a a little bit of algebra here to revive this to get three x plus one over or X to the three house. So again, we're gonna set this here and that's going to tell us that experts one is zero or excessive to negative one. Now, this would be a possible point of inflection, but this is not in our domain. So our domain, remember it was supposed to be zero to infinity. So since this is a in it, it won't be a possible point of infection. So in this case, we will have no possible points of inflection. Right? So we had wind of prime is good too. Three over X plus one over or X 23 months. And we will find that this function is Khan Kate up when y double private, strictly larger than zero. And this just happens to be our entire dough make. So com, Kate down. That's when. Why Double climate strictly lessons there and there are no intervals for this. And like we said earlier, there are going to be no inflection points since the only thing that would make our second derivative equal to zero would be acceptable, Detective One which is not in our domain. Now that we've done that, they tell us we could go ahead and start graphic. So we have our intercepts. That X is equal to zero and X is equal to three. We know at one we're going to have a minimum. What? And we know as extras to infinity the function should go to infinity. All right, so now that we have this, we can just go ahead and start connecting everything. So we're gonna start from X, is equal to zero, and we know to the right of X is equal to zero. The function is decreasing. So I'm going to decrease until I get my first point, which is going to be, except for the one which I should have a minimum there. And then after this, I need to hit the next important point, which is going to be X is equal to three. And it's just going to go up like this. So this is our nice sketch of the graph. You could go back in and possibly say what this value is for our minimum. But since we're just trying to sketch it, I think this is all the relevant information we actually