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University of North Texas

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Problem 22 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = (x - 4)\sqrt[3]{x} $

Answer

see solution

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Video Transcript

we want two steps. The CO. Why is it X minus four times the cube root of X In the structure, they give us this laundry list of that which followed. So where is gonna go one by one? Who? That was the person he suggests that we find is our domain. Now we know ex mice. War is going to be done defined for all values as well. The Cube. Ooh, uh, vets. So the domain here would just be all those numbers or negative. The next thing they suggest we buy is our intercepts. So let's go and look for the ex intercepts first. So remember, that's where our function is equal. 00 isn't too X lines for times the cube root of X. So using your product property that tells us either X minus four Is it zero or the keeper of ex busy grew zero So excessively before or ecstasy. And we don't even need to go find our Why intercept. Since when X is equal to zero, that would give us are so are y intercept will just be the origin. Next thing we can record is symmetry. So this isn't one of the functions, but it doesn't have one of the functions known for being periodic, so we could just go ahead and see if it's even or odd. See, that's so much about the origin or B Y access, so I'm plugging in negative. X will get negative X minus four times the cube root of negative X. So since I'm isn't odd degree root weaken factor that negative out and distributed toe Negative X minus four approaches the X plus four times the cube root of X. Now, this year does not equal to EPA vex. Nor is it negative EPA Lex. So this tells us we have no symmetry for the function, or at least no symmetry about why access or about the origin. The next thing we want to look for is acid toots. So in this case, we're not goingto have any true acid coats in the sense of particular values will get really close to anything, but we can at least check the end behavior of this function. So let's look at Lim has X approaches it fairly of the box, so X minus four we'll go to infinity, and the cube root of X is going to go to infinity as well. So the in behavior on the right is gonna go to infinity. And now checking the behavior on well, negative X for X mice were going to go to negative infinity and the cube root of exes, but also the negative entity and then multiplying those together. You get a N behavior. It's on insider and behavior. Go something. Now I'm gonna combine the next two steps into one. So I want to find where the box is increasing, decreasing on a local Max or Ben's. We may have so to do this we need to know what wide climates. So we could go ahead and give us on another beach. So this first step I did here was just rewrite our function to make it so. It's in something we can use. The power will take the drift above. So with 4/3 ex to me, 1/3 power minus for over three times X to the negative 2/3 power. Now we can go ahead and combine this into just a fraction and doing that little bit of algebra with three times X minus one over to square roots of X. All right now, we can go ahead and sit this function equal to zero so we can find our possible Max is. And then so the only thing that's gonna make this equal deserters where the denominators are the numerous zero so x minus one is a greaser or ex. Now, something else. We also need a check is where our grid that is defied. And that's going to be where our denominator is equal to zero. So that would say the square root of X is equal to zero or X is equal to zero. So since excessive zero is in Arbil mane of our original function, X zero will be a critical point. What? This So that isn't a Maxwell. Men, we at least know, is going to have an infinite slope at that point. All right, so now let's go ahead and write down what we found. So we found our first review before Times X minus one over three x to the 2/3. We need to figure out where this function is increasing, so it will be increasing when wide climate strictly larger than zero. So that's going to be the interval. One to infinity and we know this function is going to be strictly are decreasing when why pine is strictly less than zero, which is going to be again. Infinity to zero union is a rope. I always go ahead and put those two points we found before and see if we have Max's or Menzel, So we had X physical. Zero max is a little one, so to the left of X is a good one. The function is decreasing, and between zero and one, the focus is also decreasing. And then, after exited little one function is increasing. So at X is equal to zero. We're going to have a saddle point, and we know that should have an infinite what now At X's Widow one. This is going to be a local minimum. So a minimum, by just using the first derivative test the lost up that they said we shouldn't look away. Extra crafting is defined capacity and any inflection points. So we need to know what why double crime is equal to It's a medical here and look for the second derivative. So I'm going to use this form here. Could take the derivative just because it will make it a little bit easier to do the math. So we're going to end up with for over nine X to be negative two birds and then those negatives accounts out when I take the derivative extradited to third. And I would give a overnight out X to be negative, Hi thirds and using a little bit of algebra, we could re like this to get for X plus two overline X to be five. So again, we're going to set function equals zero. And that will only occur with experts to 00 or exited with a negative too. And then we need to set our denominator equals zero because there will be a new find there and that will tell us that x the 5/3 or just excuse you get So these are two possible points of inflection since ex physical zero is in our bill made. So we have why double Prime was acquitted or ex close to over nine X to the five fruits. So we want to find where the section was complete up. So remember, concrete up is going to be one wider. We'll find a ship with larger than their and I saw this beforehand again. Just a little bit of time with you tonight to union. And this function is going to be con Kate down on white double crimes, of course. Or this would be negative to zero. So what we need to do now is go ahead and use it to possible points of inflection which were zero right, negative to first and exits. So to the left of extraordinary to destruction, we can't give up and to the right of negative to the banks of the concave down and to the right of zero, the function is going to be okay. So we have that both of these are points of inflection. All right, now, let's go ahead and start rapping. So our inner sense we have while we have the origin. So that's our one of our ex intercepted wells I want. And then to find our other one that's going to be excessive so ever to intercept, said we have no symmetry. We know the function is going to go to positive bendy for in behavior on your side. We know that X is equal zero. We're going to have a saddle point with infinite slope So I'm just gonna go ahead and call the horizontal line vertical line like that just to remind us that we need a minute. So there and at excessively, the one we're going to have a minimum. So at a physical desirable, we're also going to have a point of in election. And at exit to mega t, we should have a point of inflection as well. So what I'm gonna do is start from negative Herb as X is the perch in negative 30 and work my way to be right now to the left of exited detonated to the function Should be conch ate up. So I'm just going to start this point here, and I should be Khan came up something like this. And after this point, the function used to be con cave down until we hit. Our one toads are next one inflection, which is our origin. And we know what that point needs to have infinite slope like you do that and then come cavities want to change on the other side and it should be calm. Keep up. We hit our minimum at 01 and we're going to go try that again. So it's like the one we have a minimum, and then it's going to start increasing again until, well, just for the rest. But we need to make sure we get our next intercept that exit. So not very prettily drone. But since we're just trying to sketch it, this year gives all the information we really need. So some things you could possibly do to just make it look a little bit nicer is saying what the value here with a point of reflection, Is that acceptable to you, too, as well as what? Our minimum value actually is excessive. But I think this is sufficient enough since we just want to sketch the graph.