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University of North Texas

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Problem 6 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = x^5 - 5x $

Answer

see solution

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Video Transcript

we want to sketch the girl. Why is he doing excellently minus violets? So in this chapter, they give us this laundry list of steps we should follow if we want to Sketch, graph, have some function. So just go on and follow the steps. So the first thing they tell us to find is our domain of the function. Well, this is a polynomial, and we know Paula. No meals have a domain of all real numbers. Although numbers or negative infinity to infinity. The next thing they suggest refined is our intercepts. So let's go ahead on back. So off. Why Intercept? Actually, go ahead. And that low down here. So are why intercept will be when X is equal to zero. So why is going to be zero to the minus by time zero, which is just d'oh and then find out ex intercepts, Remember, we're going to set why you could deserve get zero is equal to X to my ass. Bye. Let's go ahead and factor this and people get zero physical too. That's excellent work minus five and using the zero product park to get X is equal zero or ex physical to plus or minus the route. Uh, five. And this year's approximately one point. I will just need to know that approximation for later. The last thing they want us to check is for symmetry, So this won't be a periodic function, but we can check to see if it's going to be even more odd. So we won't check effort. Negative X and doing that, we will get well, negative X to the power Negative minus five. Mine effects. Well, we could go ahead and pull that negative out to power. That would be negative. One to the fifth rush. Really Negative X. And then those two natives there cancel out. And now notice if the factor that negative out we get excellent with minus five x. And this here is negative. Yeah, the banks. So with this implies is that we have a odd function or symmetry about origin. Oh, or origin. What next? They suggest we find any acid is what we know. That since this is a polynomial, there should be no acid trips. But we can use similar logic. 50 are in behavior of this craft. Should be so. I'm gonna go ahead and call this looks So as X goes to infinity. Well, this is a odd function with a positive coefficient. So we know that should go to in 30 on DDE. But they're being a odd degree polynomial than it should have the opposite in behavior for negative infinity. So it would go to negative infinity as X goes to negative. So the next two steps, I'm gonna combine Minto one. And this was supposed to be looking for intervals where it increases decreases, and we're going to look for local maxes and meds. So that means we want to figure out what wide primates have said it better than zero. So taking the derivative Oh, exit. We're gonna use power rule to take the girl with the Chinese. So we're going to get bye, thanks to the fourth by this. Now we want to figure out where there are actually some really do that. Um, let's go ahead and set up a secret. You zero. So we confined our critical points so saying to seek with zero, we would divide by five and we get thanks to the fourth minus. One is equal to zero. And then that would tell us Exit is gonna be plus or minus one once we go ahead and move the one over and take the fourth of one, All right. And actually, let me go ahead and get out of this. Exited with a negative one and ex physical one. Now what we want to do is figure out where it is increasing with decreasing. So why crime would be larger than zero. So just for remedy of problem, I went ahead. And Artie soul this so this inequality will hold on negative infinity to negative one and one to infinity. So remember this here tells us it is increasing. Next, we want to find where the function is decreasing, which were being warned. Why pride? Strictly less than zero decreasing and again already one had solved this. And it should just be negative. One too one. All right, so on negative one toe one. We know our function is gonna be decreasing. So over here, I'm just gonna go ahead and write this. So it's decreasing from negative 1 to 1. And it would be increasing before negative one, an increasing after one. So that's telling us that except the one is a minimum and exited with a negative one as a maximum. Remember both of these being local. All right. Next it says we want to determine the comm parity implants of inflection. Oh, come on, cavity on inflection points. So this means we're gonna want to look a second derivative. So why double prime while this one, the derivative respect X o our first derivative, which is five x before my life. So once again to take the dribble of X reports, it's our rule. So would get five times 4 20 x to the third power, and then the driven five is just zero. And let's go ahead and submissive zero insult that really quickly. So zero. Well, that's just gonna be X is equal to zero. Now we can go ahead and do the same thing and look for where. Why double kind of Florida's zero, which would tell us our functions can keep up there and then look for worldwide about time. It's less than zero, which would be con cave down. So live double prime zero or X cubed is gonna be larger than zero on zero Betty 20 excuse, officials say, and then it will be strictly less than zero on negative entity to zero. So to the left of zero, this is calm down. And to the right, it's Kong cable. So that tells us X zero should be a inflection point of our ground. So we followed all the steps that they suggested we do before we start actually trying to graft this. So let's first go ahead and just brought down our intercepts. So we have the intercept 00 and we have. And actually, before we do that, we know that this should be an odd function from step three. So all I'm gonna do is I'm going to only graft the left side of this and by only graphical left side, I would know I would just reflect whatever I did on the left, over to the right. All right, so we will are not reflected. All right? But reflected across the origin or the line y z x. So on the left side, we would just have 10 just going to be negative for root of I And what other things on the left we have So we know from step or that are in behavior should be going too negative insanity and we know that we have a local Max at X equals negative one. So negative one will be right here. And, you know, this would be eight Max. All right, so that should be everything for the left side of this. Now, let's just go ahead and start cropping. So our first point of importance is going to be our zero here at negative work. Beautified. And then after that, we know we're going to have a maximum around here somewhere. So what, They actually do that in a different color? So we're gonna have a maximum, maybe around here, and then it keeps on going. But then the next thing were of interest is our point of infection at exit zero. So at this point, you need to make sure those boys from Concord up on paper, it should be content down, really stressed that here and then on the other side, it should start to be con cave. And now that we have this symmetry, let's just go ahead and put in the line. Why is it ex and that we're just going to reflect everything? Um, across are not wise, and we're just gonna look everything across the board, you know, So instead of being a Max moment, one is going to be amenable. So we just call it here, put our maximum, and then this is going to just go up until it crosses the X access begin like that. So you could have went ahead and plotted in all the other points. But for this side here, I'm just using the fact from number three that we know this is and odd function with symmetry about the