use-the-guidelines-of-this-section-to-sketch-the-curve-yx4-4-x

Question

Use the guidelines of this section to sketch the curve.
$$y=x^{4}-4 x$$

Andrew Madea · Accepted Answer

we&#x27;re given the equation. F event is equal to X four minus four. Then we have to sketch first. We can figure out the domain of this function and the domain of this functions and doesn&#x27;t have an irrational or square root aspects of it. It is just negative. Infinity to infinity. There are no restrictions. You can also say that it is for all real numbers. What are the intercepts we could talk about? First is the X interest. So we can set this equation equal to zero, and we can factor out minus four. So we have an X intercept already. And then for this part, we have X cubed minus 400 And that is equal to the cube root of four. So the cube root four zero. So what about X intercept? Well, if I plug in zero for X, then we get zero minus zero. So the Y intercept is also 00 or the origin. We can perform the symmetry test so we can see if it is. Even if the original equation is the same thing as if we were placed acts with negative. We have negative X to the fourth power minus four times negative. We left with to the fourth less for this is not this original equation. It is not even. So. What about the Well, if we negate the entire function, Is that the same thing as what we got from the from the even? So what happens when we negate the entire function? We get negative X to the fore plus four X And these two functions are not the same thing because this is negative for negative four. This is positive. So it is not odd as well. So there is not even not on. And we could, um the next step is asking toes, but there are no ask him to its any polynomial function next step, since we were all this is from original function. To find more information, we have to find the first derivative, which is four X cubed minus four. And we could set this equal to zero to find any minimum or maximum values. So four X cubed minus four equal to zero we solve for X, we get remove This four over here, divided by four x cubed is equal to one or X is just equal to one And if we plug this back into the original equation so we have our X value are y value is one minus four. So that&#x27;s negative. Three. Since we have this value of one, we can figure out being intervals we contest. So we have negative infinity to one and then we have one to infinity to find out if it is increasing or decreasing. So we have our we can have a test value. We can have our sign. So for this test volume, you can say that&#x27;s a zero for this one. We can just say to So if we plug in zero to our the first derivative equation yet a value of zero minus four So that is negative to get too cute is a and that&#x27;s obviously going to be positive. So it will be decreasing on this interval, decreasing and they&#x27;ll be increased on this. So we said it was increasing on one to infinity and decreasing on negative infinity to one So and next thing we can check is concave ity. So we need to figure out the second derivative one of the river of this function which is just 12 x squared. We could find points of inflection. So we set this equal to zero. But then it is just going to be X is equal to zero. As a potential points of inflection, we could figure out intervals length negative infinity 20 to figure out the concave ity of these intervals. But notice that the second derivative is 12 x squared. This function will always be greater than or equal to zero for all X. No matter what value of extra potency here, it will be greater than equal to zero. So that means the concave ity the sign of whenever test value will be positive here, and it will be positive here. So if you chose negative one for this and if we chose that, say one for here, it would be positive, and this will be positive as well. So this is actually not a point of inflection. Oh, so you can just list here No point of inflection. The last thing we have to do is actually sketch the ground. So you know a few things about this craft? No, it&#x27;s X centers up. I&#x27;ll draw it in a blue. Oh, this one is at 00 this one is at the cube root of four zero. And then we have our what looks like a a minimum because it is decreasing from here from negative infinity toe. One, 21 and you&#x27;ll be increasing. So try to draw. That would look like he thought that like this, that is our graph f of

Bobby Barnes · Answer

we want to sketch a graph of curve. Why is it X equal with minus or ex? So in this chapter, they give us this laundry list of steps. We should follow an order to graph something, so it&#x27;s just kind of go through them step by step. So the first step is they want to figure out what the domain of dysfunction. Well, this is a polynomial, and we know polynomial is have a domain of all real numbers or negative infinity. The next part is one to figure our intercepts, so intercepts. So let&#x27;s go ahead and find our Why intercept first, just because that&#x27;s a little bit quicker. So why intercept is when X equals zero? So that would give us why is equitable? Extra support is zero and then four x zero. So we get zero for why interested and now for our X intercept. Remember, he said, why equal does aerial? So you get zero is equal to X to the fourth minus for X. Let&#x27;s go ahead and pull that or or the X out, and we get X Times X huge minus or so This tells us either X is equal to zero or X visible to the cube root of four. And this year is approximate the one point by. All right. Next, I did not go next. We want to see if there&#x27;s any symmetry. Intergraph, which is we want to look at of negative X and so doing this year. And so by EPA, Becks, I just mean our original function here. Why? So we can go ahead now and plug in negative X into here and doing that so plugging negative X and X support will just be extra aboard and then plug in negative x in for X with the negative X and that it would become plus four X. Now, this here does not equal to FX, Nor is it negative FX. So what this tells us is no symmetry court, so we got the first little bit done. Now, the next thing they suggest we buy is any acid trips. So we know since this is a polynomial, there should be no acid tips. But we can at least follow the similar steps to figure out it. What kind of in behavior we should have for this. So what we&#x27;re going to look at is the limit as X approaches Infinity of affects. Well, this is a court IQ function, and we know those kind of look like x squared. So since I was a positive coefficient, we know this is going to go to infinity. And likewise, we know what we have been. Even highest power for a polynomial. We will have the same. So the limit as X approaches negative penny of after Max also goes to the next thing they want. Supply is where this function is increasing and decreasing, so increasing slash decreasing intervals. So that means we want to find why pride. So why prime is going to be so be derivative with respect to X Oh X to the fourth minus war it&#x27;s and to take the derivative of both of these weeds powerful. So first we will get or X now Q. Because we need instruct all that power and then the derivative before X a jest or so. Let&#x27;s go ahead and factor that four out. So four X cubed, minus one. So if we were to just kind of look at what X cubed minus one looks like we know it&#x27;ll look something along the lines of this now it might kind of curve, but at least we know it&#x27;ll look something like It&#x27;ll fly an island in, Go back again so well that toeses this here. So why prime will be less than zero win. So let&#x27;s sexually first step that equals zero to figure out. But that should be so if we said it equals zero that just tell those X should be one So too the left of that point We know the function would be negative. So this will be less than zero when we&#x27;re on interval. Negative Infinity Q zero. And to remember this here tells us that the function is decreasing and when wide prime is strictly larger than zero. So this should be zero should be infinity too. The value What? So because of that our necks, then it would be one to infinity. And this here is our interval where it is increasing. Oh, so normally you would just set this equal to it are put into a inequality to figure out where it&#x27;s bigger than equal to zero. But we&#x27;re just gonna take the cheap way out so you don&#x27;t have to do more algorithm with you all right. The next step is to find our local Max is slash men&#x27;s and what we need to find those are critical points and we already found that are critical. Point is exiting the one because we want to find a wide prime is zero and we already know that sex is equal One last step. Now we can use the fact that we know on the interval of negative finito won, the function is decreasing and then it&#x27;s increasing after. So this tells us we have a local men at exit one. The next step is to look for con cavity and any inflection points we might have. So that means we need to look at the second derivative. So why double price? So we&#x27;re gonna go ahead and take more X cubed minus four and take the derivative of that that would be derivative with respect to X or excuse me in this war, so would go ahead to use power would take the derivative of X cubes, love the three x squared. We have well X squared and in the derivative of or we&#x27;ll just be zero. Now notice that this function here, we&#x27;ll always be greater than or equal to zero. And because of that, we know this wall be con cave up so con a Oh, and also there will be no inflection points Now that we have all this information, this was the last thing they thought was important so we could go ahead and try to grab our option. So let&#x27;s go ahead and first just locked down our exit. My inner sense. So we have our X intercept being or a wire set being 00 And we have our X intercept 10 and the other one is about one point. Just go ahead and put you grew Oh, four. All right, so we have those intercepts style, So there&#x27;s no symmetry our ass in tokes we know Well, there were no acid totes, but we do have the end behavior being on both sides, going to infinity. We know at X is equal to one. We have a local men, so now we can just go ahead and connect our graphs and everything. So it&#x27;s going to come from infinity, and the first point it needs to hit is due. And then, well, we said they were supposed to be a local men at 11 So it needs to curve up again like this, and then the next point need you need to make sure we get would be our other X intercept And then this is just gonna go off until like this. Now it&#x27;s not drawing very well, but at least you kind of get the idea. That should be more smooth. But unfortunately for me, I was not an artist. Now the only thing I might also say he should find is what value is here. Ah, local men. But I really don&#x27;t think it&#x27;s too important since we&#x27;re just trying to sketch graph with it. But this is all I think we really need to show for our graph. Just maybe draw a little bit.

Bobby Barnes · Answer

we want to sketch the graph of why is he going to x over expert mind sport Now in the structure they give us this laundry list of steps we should follow to sketch a graph. And one of the things we should always do for any rational function is to go ahead and factor in your area. The Dominator, Since we need to be ableto find any kind of holes or vertical ascent Oops, as well as our ex intercepts. So memory doesn&#x27;t factor, since just but the denominator is different square. So it&#x27;s gonna be X plus two X minus two and X plus two all over X. All right now, we can&#x27;t cancel anything else. Let&#x27;s go to the first step, which is our domain. Well, we know we need to make sure that our denominator does not equal to zero. So that&#x27;s gonna be an exit with the two and negative too. So the domain is gonna be negative. Infinity too Negative to union. Negative too. 22 Union to To infinity. Let me go ahead and drop this down. What? The next thing they tell us that the court is our intercepts. So let&#x27;s go ahead and find our X intercept First X intercept. Well, that&#x27;s going to be where the function is equal to zero. Or why is it so? Zero is equal to X over X squared minus four. And I&#x27;m not factoring that it on there because remember, no matter what we plug into the dominator, you never make this equals zero. So Soling for zero, we would just get X is equal to zero and since exited desirable. That would also tell us our Winer said it would be at the same point. So bye intercept is also going to just be why is zero? So the origin is our only interest. The next thing they suggest we find is any kind of symmetry we might have. So national functions aren&#x27;t really known for being periodic. So what we want to check him said, is whether the function is even or odd. And we do that, remember, by looking at of negative X, so we&#x27;ll get the negative X we get negative X and negative. X squared is just going to be ex squared minus four. Now notice that this here is equal to negative of necks. So this here implies that we have an odd function. So we have symmetry about our origin. All right, the next thing we need to chart core is our ass in Toots. Now we know that the limit as X approaches. So plus and minus infinity of our function at the backs, which is really just why so since our denominator has the larger power going back to dr 2.6, this tells us we&#x27;ll have a war zone toss until at, why? Easy, which is zero. All right, now we need to figure out our vertical acid vote since we just found our horizontal using that information. So the limit as X approaches. So we said our Berta classic. Oh, it&#x27;s going to be Where are the dominators? Zero. So X is equal to an X equals negative. So start with negative, too. So negative, too. From the right of Ethel bex. So let&#x27;s go ahead and plug that in. This could be negative too. From the right. Oh, negative too. From the right. Minus two times. Negative too. From the right. Plus two. Uh, let me go screw that over a little bit now. Negative, too. From the right is still going to be negative if I subtract so negative too from the right is gonna be negative again. And then if I subtract something that&#x27;s still gonna be negative and then if I do negative too. To the right, that will be something like negative 1.99 If I add that to two would be positive, so is gonna be overall positive. So that tells us that this should go to positive infinity. And we don&#x27;t even need to do the same steps are looking for as we approach it from the left because we know both. These are Ligier terms. That means they should have opposite behavior on either side. So just to save a little bit of time, we know that this here should go too negative, and Bennett and you could go to the same logic that we didn&#x27;t know before, but just to save a little bit of time. Now, the next thing we want to do is look at our other interesting, which is going to our other verte class, which is gonna be as X approaches to. So let&#x27;s look for right again. Let&#x27;s start so it&#x27;s ready to from the right over two from the right, minus two times two from the right, plus two and using a similar logic that we did before. So the new mayor is going to be positive. Slightly to the right. We&#x27;ll still be a positive number now. Something slightly to the right of two will be like 2.1 and subtracting, too. From that well, that would still be positive and tune from the right will be all positive number. So addict, something possibly possibly as well. So this here will approach positivity and using similar logic that we did before. We can find out that as we approach to formula, this should be negative. Infinity already. The next step is we want to find our intervals of where this function is increasing and decreasing and any kind of local Max&#x27;s ornaments. So this is That&#x27;s five and six, and I think they should be combined into one step. So we want to find what, why prime ms. So let&#x27;s go ahead and do that on another page. So if y is equal to X over X squared my sport now take the derivative of this. We&#x27;re going to need to use questionable. So question rule is low d high minus high d low all over the square. What&#x27;s below? And these just represent derivative. So going in filling that out, we will get the following expression and X squared minus for And then we need to square that now the derivative of X is going to be one, and the derivative of X squared should be to excusing the power rule and in the derivative of, um, negative forward, just be zero. So that should be two X and we can go ahead and rewrite this. Ah X squared minus four minus two x squared All over X squared minus four, I swear. Then, in the numerator, shove negative X squared minus four. Unless factor that negative out. And I will leave us with X squared plus four in the numerator over X squared minus four squared. Now notice that expert plus four will always be strictly greater than zero. And expert minus four squared should also always be greater than or equal to zero. So that means since we had that negative out here, this function will always be less than zero or decreasing. So go ahead and use that. So let&#x27;s go ahead and write. What? Why? Prime is real fast. Well, wide prime was negative, X squared plus four over X weird minus four square. And we already said that this year is going to be strictly less than zero. So our interval of decreasing will be everywhere that the function is just defied. So it&#x27;s always decreasing since why Prime Mister Crewe? Us these air. So it would just be negative infinity to make it up to union. Negative too, to do union to to infinity. All right. And we have no critical values for this since the numerator can ever equal zero. So the function can never actually equals zero and are derivative is undefined at X is equal to two and negative too. But those are not in our domain, so we don&#x27;t need to check those for critical ones. All right, The next step is we need to find our con cavity, so call on cavity and any kind of inflection points. So that means we need to figure out what? Why double prime Ms actually go ahead and move this suicide over here instead. So let&#x27;s just go ahead and little barrier there. So, like I&#x27;m saying, we need to go find what? Why? Double promise first. So let&#x27;s go over to hear again and start taking the derivative. So why double prime? So we can factor that negative out front and then just apply bullshit rule again. So we are going to do low d high, so X squared minus for squared times, the derivative of what&#x27;s above, which is X square plus four minus. Then the opposite order. So x squared plus four times the derivative with respect to X of what we have in the denominator x squared minus for squared and then all over what we have in the denominator squared. So X squared minus four squared But squaring that would give us to the power. And so don&#x27;t forget this You negative out here now, taking the derivative ot to these well expert plus four is just going to be two x and to take the drilled of X squared minus sportswear do enough to use Powerball and chain will. So the derivative of the outside, his first going to be two x squared minus four and then the derivative on the inside is going to be two x part. And just so we don&#x27;t have to do all that algebra when we simplify this, we should be left with two x x squared plus 12 over X squared minus four. Keep. So let&#x27;s go ahead and use that. Actually, before we do that, let&#x27;s figure out where this function is equal to zero. So again the denominator can never eat with zero, so we don&#x27;t need to worry about that. So we just need to worry about the numerator. Well, X squared plus 12 can ever equal to zero. And we know two x can. So we get to X zero or exes just is there. So this is one possible point of inflection. All right, so it was the hand right down that derivative or second, Really, we just found you&#x27;re going to be two x x squared lost well over X, where minus four Q. And just for the sake of brevity, to find where this option strictly larger than zero, I already went ahead and solved. That is, well, why Double prime greater than zero tells us what the function is comfortable and that&#x27;s going to be on negative, too. 20 Union too. Then, fans, Why double prime theatrically less than zero. Well, this stuff this is going to be conch aid down. And this is just the rest of the using interval from our domain. So negative many to negative to union with zero toe too. So we found that ecstasy. Coke zero is a possible point of inflection. So let&#x27;s go ahead and see if we have a change in Kong cabin here. So to the right of zero, we will be conch eight down and to the left of zero, We&#x27;re going to be con cave. So this here tells us we do have a point of inflection at X is equal to zero. All right, now, let&#x27;s go ahead and actually grab this function. So we know we have our intercept at only the origin. We know the function is odd, so it should be symmetric about the origin. And now we can go ahead and plot our ass in toots. So we know we haven&#x27;t asked. Toe X is equal to negative too, and X is equal to two. So excited, too accident negative too. And we have a horizontal ask himto at why is equal to zero. Um, so we have no local men&#x27;s or maxims from 0.5 and six, but we know at zero at X is equal zero wish of a change in con cabinet. What now? For horizontal Herbert glasses tips. So let&#x27;s start there to the right of negative too. We said that this should be going to positive infinity until the left of negative, too. We should be going too negative infinity. And we also said to the right of two we should be going to positivity until after to wish you were going to negative infinity. Now we can go ahead and start grabbing, so I&#x27;m just gonna start in this middle section, so we&#x27;re gonna start to the right of negative too, and we need to go to our origin. And at that point, it&#x27;s one to switch con cavity. So to the left of this, it&#x27;s complicated. But to the right, is Kong cave down now to the right of two? Well, we have no points of interest, so we&#x27;re just going to go until we hit our horizontal awesome too. And we will need to approach it from above as we can see there and now you can go ahead and go to the left of negative two. We have no points of interest after that. So we&#x27;re just going to go toe was awful, Jacinto and approach it from below, so you could maybe plot some actual numbers to make it look a little bit prettier. But since we&#x27;re just sketching it, I think this is a good place to stop.

Bobby Barnes · Answer

we want to sketch the curve. Why is he do X times X minus or cute? Now, in this chapter to give us a laundry list? Oh, steps we should follow in order to graft something. So just go ahead and follow them step by step. So the first thing they suggest that we do is to determine our domain. So we have a polynomial here, factored polynomial, and we know that the domain of the polynomial is just all real numbers or negative infinity. The next thing they say we should find is our intercepts. So are why intercept is going to be when X is equal to zero. So we&#x27;re gonna have Why is he going to zero zero Bias for and that&#x27;s just zero are x intersects are going to be when why is equal to zero sort of. What had zero is equal to right. Zero physical too X times x minus for cute and we can use his their product property. The X is equal to zero, and X is equal to the next thing that you suggest we do is to look for symmetry. Now we know that this is not gonna be a periodic function, but we can check to see if of negative X is upper Becks for negative ethics. So plug in end that negative there would get negative X negative X minus four Huge. Now we can factor that negative x out and had become negative. One cube. So negative one cube is just negative one and the negative outfront counsel&#x27;s weaken X X close or cute. Now, this does not equal to FX, nor that negative the next. So what this tells us is we have no symmetry. The next thing is, we want to look for acid toots. Well, we know this shouldn&#x27;t have any acid tubs due to it being a partner, but we can use that same logic to look at the end behavior. But let&#x27;s just go ahead and do that. So the limit as experts, infinity of epidemics. Well, in this case, I mean up about being our function over here. Well, this is essentially why to the fourth power. Um, just because I would be the largest degree and the official would be just one, so we know that even degree polynomial ls with positive politicians go to infinity and we know that the in behavior on both sides of even degree polynomial ls will be the same. So we get that are in behavior will be infinity and infinity. Now, the next two steps I&#x27;m gonna go an emergent of one and it&#x27;s finding our intervals of where the function is increasing. We&#x27;re decreasing, and also we want to look for max slash mittens. So let&#x27;s just go ahead and do this on another page toe, save some space. So we have Why is he going to x times X minus four? So we have two functions being multiplied together, So we&#x27;re gonna need to apply the product. Remember, product will says first function, take the derivative of the second and multiply them together, and then we&#x27;re gonna add them in the opposite order here. So first we know the derivative of X is just gonna be one. So it&#x27;s just going to simplify that, and then to take the derivative of x minus for Cuba would have to supply chain. So General says, take the derivative over outside puncture, which in this case is gonna be excused. So that will be powerful. So three X linus uncle minus or squared and then we need to take the derivative of our inside function, which is X minus. Or now the driver of ex my score is going to be one because derivative of X is one derivative 40 not one. And then here we just have X minus four. All right, now, let&#x27;s go ahead and factor out. Ex. My sport squared and doing that will leave us with three x plus X minus four. So on the inside here we get for X minus four, so we can factor of four hours. Well, four x nice four square and then x minus one. All right, so we have y prime. And if we were to Septus equal to zero, we would get that are critical points. We&#x27;re going to be X is a good one, and X is equal to or so we&#x27;ll need these critical points in a minute. We can look at these issues. People being back, sir. Men&#x27;s all right. So we now know that why prime is equal to or Times X minus four squared and X minus one. All right, and what we want to do to figure out where it&#x27;s increasing or decreasing. It&#x27;s the girl worldwide Prime is strictly loved, with zero and strictly less than zero. So for increasing, we want wide prime to be strictly greater than zero. And just to say both some time, I went ahead and did this earlier. Otherwise, there&#x27;s video a little bit too long. So this function is increasing on 124 and then or two and we can figure out where this function is decreasing by looking at where wide private Shipley lessons there I did once again just for the brevity of the video headed Did this already the negative to one. So let&#x27;s go ahead and put those critical points we found before down here. So we had that X is equal to one and X is equal to. So now we&#x27;re gonna imply the first riveted test. All right, so to the left of X is equal to one. We know the function will be decreasing in Texas with one, and then after excessive little one, we know the function is increasing, so it decreases in tow. One increases after, so we know this should be a local men. So I&#x27;m saying backs in men, but I should be saying local maximums and now exit before, so it&#x27;s increasing in before and after four. It&#x27;s also increasing. So this is not a backs or Ben. It would be a saddle want. All right, so we have our max and our men&#x27;s tackled. The next thing we want to do is check or contact in and inflection points Okan cavity slashed and collection points. So that means we want to look at Why Double prime and let&#x27;s just go ahead and go back to that last page of Cigar. What? Why Supply Double Prime is going to equal to well, the derivative of two functions being multiplied together again, Remember, since or is a constant, we can just go ahead and factor that out, since the derivative is linear or squared and X minus one. So we&#x27;re gonna apply product cool once again. So we&#x27;re gonna have X minus four. Where times the derivative of X minus one plus. Explain this one time too derivative with respect, X um X minus four squid. So you get explain its or square. So, just like before, the derivative of X plants, one is just want to be, uh, one, and you take the derivative of X minus or square, going out to apply change world and, uh, chain rule. So first used Probable two. That&#x27;s minus four now for the first power. And then we need to take the derivative of our inside function, which is X minus four. And again, the derivative of X Vice Lord is just going to be one. Or let&#x27;s go ahead and put this up a little bit. Um, so first, let&#x27;s go ahead and background ex my sport from this X minus four. So that would be fine. X minus. For there had been a plus two x minus two before X miles more simplify the inside. Here we would end up with three X minus six and like a factor a three out of there to get four times x minus four that a fact about that threes. There should be 12 out front here and exploits for and X minus two. So if we accept this equal to zero, we would get that are points X is equal to and X is he good for? So we could use these in a moment to look for points of an election. So let&#x27;s go ahead over here and right down with our second orbit of days, which should be, well times X minus four times. That&#x27;s minus. And to figure out where to calm. Keep up a calm down. We&#x27;re going to need to look at where Why Double kind is strictly Larkin narrow and strictly lesson. So Khan taped up is when, Why, that will find a strictly larger than Europe. And then, once again, just for the brevity of the problem, I went ahead and solved this already. So this is gonna be evident. Did you union with or to infinity? And over here for down this will be one lie double promise strictly lessons era. And we would be from 2 to 4 Now are two possible points of inflection are gonna be X is equal to and X is equal before So let&#x27;s go ahead and see so to the left of X is equal to two. We&#x27;re going to be con cade down to the right of two. We&#x27;re going to be, uh, to the left of two were actually on this interval here, So we should be Khan paid up and then to the right of two were on the other interval for being down. All right, so this here would tell us we have, ah, inflection point because cavity changes at accessibility. But then at exit Eagle four after four, it will be Kong cape up. So both of these are points of and lecture. So that was everything they suggested we do before we actually start to sketch this. All right, so let&#x27;s go ahead and first just locked down our interest. Since so we have our Y intercept would be 00 Our X intercept should be zero and executed 41234 So there was no symmetry in the problem. Our Assam totes will no acid trips. But we at least know the in behavior of this craft is going to be going to infinity on each side. We know we&#x27;re going to have a local men at Exit one. So there&#x27;s a man around here somewhere and we could figure out whether that needs to be above the X axis or below the ex Access access in a second and at X is equal to so maybe actually above the one I&#x27;ll quit men and then at two, and or we will have inflection points, so I p Hi, Pete. So look ahead and start with this. So we&#x27;re going to start from infinity and we&#x27;re gonna go down until we have our first point. We&#x27;re just going to be 00 and then at this point, our next important point is our minimum, that ecstasy with one. So we know it&#x27;s going to curve up something like this here and at exit to we&#x27;re going to have a change in inflection. So it starts to go from conch Yup, to concrete, down and at exit before we have the same kind of slowly comes in and does something like that. They&#x27;re so maybe that curve at Exeter before should be a little bit Maur intense, something kind of light This so it&#x27;s a kind of flatten it out a little bit more, but they used everything over here. We used our inflection points. We can see the con cavity where it&#x27;s increasing and decreasing. And if I was able to draw a little bit better, that might look a little bit prettier. Um, something else you could do is go in and maybe say what value this occurs that but as far as I&#x27;m concerned, since we&#x27;re just trying to Oh, as well as this inflexion point there. But as far as I&#x27;m concerned, just cause we&#x27;re trying to sketch it, I don&#x27;t think these numbers matter all that much.

Bobby Barnes · Answer

we want to sketch the curve. Why is it for my sex where to be the power? So in this shocker, they give us this laundry list of steps we should follow whenever we want to grab something. So we&#x27;re gonna follow those steps. So the first thing they tell us is to determine our domain. Well, this here is a somewhat factored polynomial. So we know our domain is going to be all build numbers. The next thing they tell us to do is to find our intercepts. So intercepts and let&#x27;s go ahead and find our why intercept first. So why intercept? This is when X is equal to zero. So it&#x27;s gonna be why is it too? Or minus zero squared, which would just be zero. And so it before to the Yeah. All right. Um, so I really don&#x27;t care what this number is. I just care that it&#x27;s possible Negative. So we&#x27;re just gonna leave in this sport, but you can rewrite it if you want. Next. Let&#x27;s go ahead and find our ex intercepts. So now this is when Why is it zero? So we&#x27;re going to have zero is equal to or minus squared to the fifth, Will. This implies that we really just have 00 to 4 minus X squared, which is X is equal to plus or minus two. And the last thing is, they want us to find symmetry. I&#x27;m going with him. Move these over a little bit, but I lost my zero along the way. Next. So Step three should be symmetry, so we don&#x27;t polynomial. They&#x27;re not gonna be periodic, but we can look to see if it&#x27;s even on persimmon tree about the wire. Ex access. I mean, the Y axis were the order. So plug in this inn, we&#x27;re gonna get four minus a negative X square to the foot. It&#x27;s wearing the negative extra sees a sex. So we get four minus x squared to the phone, which is EPA, Becks, and by epa, Lex. I mean, the function we&#x27;re working with up here, so this implies that this is an even function. So you have symmetry across B Y access. The next thing, they want us to find his ass. So we know that polynomial should not have any acid. Oops, but we can go ahead and look at the end behavior with the same idea. So the limit as X approaches and very of pepper vex Well, this would be really so I&#x27;m just kind of break. This is for very large value is going to be cool, too. Negative x to the 10. So we know it is even degree polynomial the negative. So efficient. So this year should go too negative. Infinity. And since it&#x27;s and even degree polynomial whatever behaviors on one side, it&#x27;ll be on the other. So this should also be negative. The next thing we want to find is intervals of and I&#x27;m gonna combine this Step six Awesome. We want to find where the function is increasing, slash decreasing and any local Max Cashman&#x27;s that we have. So that means we&#x27;re gonna first need to find what wide prime it&#x27;s. So let&#x27;s go ahead and do that on another page, since this will take a little bit of work. So why is equal to or minus X squared to the left? Well, the first thing I&#x27;m gonna do is just factor out a negative from here. So I wanna write negative X squared minus four to the fifth. Remember, we can factor that out since it&#x27;s an odd power, all right? And I just want to do this because it&#x27;ll help me keep things in place when I&#x27;m taking derivatives and stuff. So the first derivative will be, Well, we&#x27;re gonna need to use the Powerball and changeable for this. So first would have squared minus four to now the fourth power, and then we need to multiply a five out front. Then we need to take the derivative over inside function, which is going to be X squared minus sport. And we know the derivative of this here is just going to be two x, so we end up with negative 10 x x squared minus four to the fourth. Now, we&#x27;re going to go ahead and set up a secret zero because we need to know that critical points of this Help us find our local maxes and local mittens if we do happen. So this would imply that there X is equal to zero or X squared. Minus four is a good a zero. And once again, this is gonna be X is equal to plus or minus two so critical place. They&#x27;re zero and plus a nice to you. All right. So now let&#x27;s go ahead and write down. What wide promise. So, Floyd prime is going to be negative. Jim Specs, times X squared minus four to the fourth power. Now we can go ahead and determined where this is going to the men that makes you write this right, So increasing. Which will be where? Why, Prime, it&#x27;s strictly larger than zero. So I just went ahead and already solved for this. Just thio cut down the length of the video a little bit. That would take too long to do this. So we would have that this is strictly grade zero on the interval of negative infinity to negative too. And then union negative too. 20 And we know this is going to be decreasing when Why Prime is strictly less than zero. And this would just be the rest of the interval. Minus are critical points. So it would be zero. Did you union to To Infinity. Now let&#x27;s go Then plop down those critical points we have. So we had negative to zero and two and we know this will be increasing on negative negative too and negative. 2 to 0. And we know it&#x27;s going to be decreasing on 0 to 2 and two to infinity. So that tells us Negative, too, is a saddle point by the first derivative test two is also a saddle point by the first derivative tests, since we do not have a change in the derivatives. But we get that zero here, since it&#x27;s going from increasing to decreasing. Should be a local backs, all right? And then the last thing they suggest we find is the con cavity of this and our inflection points. So we&#x27;ll need to know what the second derivative of this function will be. So let&#x27;s go over here and do that also. So we found that. Why prime is this right here? So we want to take the derivative of that and we&#x27;ll do that by using the product. Cool. So why double prime is going to be and I&#x27;m gonna factor that negative 10 out front here, remember, product will says, keep the person you&#x27;re multiplying and then multiply by the derivative of the second function. So X squared, minus four forth. And then we&#x27;re going to add to this where they&#x27;re switched so X squared, minus war before and then the derivative with respect to X of the other culture, which is X. So now we already know the derivative of X is just going to be one. And to take the derivative of expert minus four to the fourth, it&#x27;ll be just like before where we had to use changeable. So the board is gonna come out front, X squared minus four and ours to the third power. Then you need to take the derivative of our inside function so expired minus four. And then over here, we&#x27;ll just have X word minus four to the fourth Power. Now the derivative of X Square is gonna be to expect power will get and then the derivative for just zero, since it&#x27;s a constant. So let&#x27;s go ahead and clean that up a little bit. This could be negative. 10 eight x squared, X squared, minus or thirds waas x squared minus four to the port. So we could go ahead and factor out this X squared minus four to the third. And doing that will leave us behind with eight x squared plus X squared minus four, which could be simplified down further. Two nine X squared, minus So now if I were just setting this equal to zero will notice We have two perfect squares here, and we already know that this here will get us X is equal to plus applies to. And this here will give us X is equal to closer minus two. So factor that since they&#x27;re perfect squares I mean the difference of squares, not perfect squares. And it should just be plus or minus. Those answers are so we&#x27;ll need to know that there as well. All right, so let&#x27;s go ahead and write down our second derivative, so it should be negative. Tip X squared by this four cubed and nine X squared minus four. Now, just like we did before, we want to figure out where this would be con cave up, calm down so we can figure out our inflection points. So Kong Kate up is gonna be when? Why, double private? Strictly greater than zero. And then, just like before, just for the sake of brevity, we will, um, I&#x27;ll just say where this occurs so we don&#x27;t have to go through all the tedious algebra. So this year will be on negative too, to negative 2/3 and also 2/3 22 And then the remaining interval or inter bowls will just be where it&#x27;s Kong came down. So negative. Infinity two magnitude union negative 2/3 two, 2/3 and two to infinity. Now we can go ahead and use this to figure out our once of inflection. So before we found negative too negative. 2/3 2/3 and two. Now we know this is Kong. Keep up on negative too, to negative 2/3 and 2/3 to 2 and it will be conquered down on the rest of these intervals. So negative 2 to 2 has a switching call cavity. This is inflection point the same thing for negative 2/3. Same thing for 2/3 and same thing for so all for these are points of inflection. All right, now that we have this done, it tells us we could go ahead and finally sketches. The person we should probably plot is our intercepts, and we&#x27;re going to have zero negative too. And two, or are so 0.0, we&#x27;re going to settle our why Intercept actually was supposed to be up here at or to the fifth and then the ex intercepts are negative, too. All right, so we have that. And by number three, we know this is asymmetric function. So if we just plot the left hand side of this, then we could just reflect that graph across the Y access. All right, so on the left, we have are in behavior going to negative 20 from part for, so we know we need to start down here. And what other important points do we have? What? Negative, too. We have a saddle point. So that means it should fly and helped right here, and we know it. Zero. It will, um, be a max. And we also have inflection points that negative 2/3 so negative 2/3 will need an inflection point. So let&#x27;s go ahead and start wrapping this now. So you started negative, Penny. And the first point of interest is going to be negative, too. And like I was saying, it needs to flatten out here, since this is a saddle point, and it&#x27;s also a point of where our cavity will change. And then we hit negative 2/3 and the capacity should start to change as well. We hit our maximum here, zero from park five and six. And now all we need to do is reflect this graph across the Y access due to seven tree. So, bye, really? You can just go ahead and starts the same thing you did before. So it flattens out here and then goes down too. Thank you. So you could go ahead and be a little bit more accurate with this if you want. But since we&#x27;re just trying to sketch it, I don&#x27;t think we really need to add all these numbers and stuff. But you could always go back and possibly figure out Like what this actual value here is for the changes on cavity. Same thing on the other side. But I don&#x27;t think it really matters all that much since we just want to sketch it.

Bobby Barnes · Answer

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&#x27;s go and look for the ex intercepts first. So remember, that&#x27;s where our function is equal. 00 isn&#x27;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&#x27;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&#x27;t one of the functions, but it doesn&#x27;t have one of the functions known for being periodic, so we could just go ahead and see if it&#x27;s even or odd. See, that&#x27;s so much about the origin or B Y access, so I&#x27;m plugging in negative. X will get negative X minus four times the cube root of negative X. So since I&#x27;m isn&#x27;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&#x27;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&#x27;s look at Lim has X approaches it fairly of the box, so X minus four we&#x27;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&#x27;s on insider and behavior. Go something. Now I&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;t a Maxwell. Men, we at least know, is going to have an infinite slope at that point. All right, so now let&#x27;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&#x27;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&#x27;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&#x27;re going to have a saddle point, and we know that should have an infinite what now At X&#x27;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&#x27;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&#x27;s a medical here and look for the second derivative. So I&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s go ahead and start rapping. So our inner sense we have while we have the origin. So that&#x27;s our one of our ex intercepted wells I want. And then to find our other one that&#x27;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&#x27;re going to have a saddle point with infinite slope So I&#x27;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&#x27;re going to have a minimum. So at a physical desirable, we&#x27;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&#x27;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&#x27;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&#x27;re going to go try that again. So it&#x27;s like the one we have a minimum, and then it&#x27;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&#x27;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.

Problem

Use the guidelines of this section to sketch the …

Problem 3 Medium Difficulty

Use the guidelines of this section to sketch the curve.
$$y=x^{4}-4 x$$

Answer

$(-\infty, \infty)$

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James Stewart

Essential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$(-\infty, \infty)$

Essential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

James Stewart

Essential Calculus Early Transcendentals