find-a-cubic-function-ya-x3b-x2c-xd-whose-graph-has-horizontal-tangents-at-the-points-26-and-20

Name: find a cubic function ya x3b x2c xd whose graph has horizontal tangents at the points 26 and 20
Uploaded: 2020-05-17
Duration: 7 min 1 s
Channel: Numerade
Description: Find a cubic function $y=a x^{3}+b x^{2}+c x+d$ whose graph has horizontal tangents at the points $(-2,6)$ and $(2,0)$ .

Question

Find a cubic function $y=a x^{3}+b x^{2}+c x+d$ whose graph has horizontal tangents at the points $(-2,6)$ and $(2,0)$ .

Wesley Hines · Accepted Answer

given the horizontal tangents to zero and negative to six. We want to come up with a cubic function that is a X cubed plus B X squared plus C explicit E that will give us the tours on tangents. Now, one way to solve this is through systems of equations because we have to solve for four different variables at the same time. That is a, B, C and D, nor for systems of equations to work here, though, we&#x27;re going to need at least four equations that contain a, B, C and D as you need at least one equation her variable you want to solve for. So one way we can get two of our initial equations is with the horizontal and tangents were given. That is that we know that when we plug in to we get out zero. When we plug in negative two, we get out. Six. So doing this with our function. Plug in to you. Okay. Eight a plus four b was to see Rusty and I would be equal to zero and then likewise was negative. Two. We will get negative. Eight a plus four B minus two C, Steve and that will be equal to six. So now we have two equations here, but we need at least four, and one of the cool things we can do with these horizontal tangents is one of the defining features of a horizontal tension. Is they happened when the derivative of your function is equal to zero? So if we derive our function and said equal to zero, we can use thes same horizontal tangents to come up with two more equations and so driving or function using the power rule on all four terms will give us three a x squared plus to be X plus C that is equal to the derivative of our function. Now, if we just take this Dr Function and plug in to a negative to, that would give us two of our other functions. And these are both equal to zero because, as we said earlier, one of the defining parts of a horizontal tangent is that the drift of your function should equal zero. At that point. Doing this will give us 12 a plus four B plus c. She put zero and doing this with negative too. Give us the same thing, but with a negative for beauty instead of positive for B now to solve. For these equations. What we Cano&#x27;s with the third and the 4th 1 is that they both have a 12 day and they both have a seat. So if we subtract thes two from each other, that is to say, 12. A place for people C minus 12 a minus four B plus c. Doing so will cancel. Er is zero a. Make our bees into a positive four B minus negative for B, which would be a plot positive giving us baby plus C minus c. You should get a serious C is equal to zero. And from here, if we divide Ates will 0/8 is going to give us. B is equal to zero. And now that we know that musical zero, we can thin up a lot of our equations because all of them technically no longer have a B term as it&#x27;s equal to zero. If we announce your attention towards the 1st 2 equations, we&#x27;ll see that we have something similar except with eight A and negative eight a and to see and negative to see. And that is they&#x27;re the exact same constant terms in front of each variable. The only difference is that the signs are different, and this tells us that if we add the two functions together that those terms should cancel in doing so. If we take a plus negative eight, they will give us here away, for bees are both zero, so zero plus year we&#x27;ll get. So give a syrup two C minus two C, giving zero c and then do you? Plus do you giving to D just equal to zero plus six six. From here, we can just divide a two out from both sides, and this will give us de is equal to three now. From here, we solved a sulfur and see, but there&#x27;s no nice way to just add or subtract one of the lines from another to simplify the equation to solve. But will, we can dio is we can set one of the variables equal to another. So, for example, if we take that third equation 12 a plus C is equal to zero weekends, attract 12 a from one side and has to give a C is equal to negative 12 and now we can pluck in this negative 12 a instead of C for a different equation. So, for example, review of the first equation we&#x27;ll get eight a plus two times. See? Well, as we said, See is you got a negative 12 a. So I&#x27;m going to substitute that out and then let&#x27;s do you is equal to zero Simplifying. This will give us negative 16 a plus three as D is equal to three, and that&#x27;s equal to zero. And then from here, solving out we&#x27;ll give three is you go to 16 A, which will give a is equal to 3/16. And now that we have a B and D soft for, we can substitute those three variables and into any of our equations to solve for C, and we can even substitute it in here to solve. Received and doing so will get us see is equal to negative 9/4. And as we said earlier, nor to solve this problem, we just need to come up with a value for a, B, C and D in order to solve this problem. Now that we have our said values, we just plugged them into our function. We came up with represented cubic function, and that&#x27;s say that 3/16 X cubed plus zero X squared, minus 9/4 X plus three will give us a function that has horizontal tangents at 20 and negative to six.

Clarissa Noh · Answer

he had slower. So when you right here. So we have. Why is equal to a X cute? Must be X square plus feet x plus D you differentiate this to get three a X Square plus two B X? Let&#x27;s see. We&#x27;re gonna set up our equations from our given points, so we have negative to calm a six when we plug it into the original equation. We got six in our line ISS negative. Eat a That&#x27;s four p minus two C plus T is equal to six. Then we have 0.2 comma zero. When we plug it into our original equation, we good cereal. Yeah, a Ame plus four B plus two C plus D is equal to zero. Where are derivatives since there&#x27;s horizontal tangents, the derivative for positive and negative, too, is equal to zero, so we go for negative to calm a six. You get our derivative to be equal to zero well, a minus four B plus C. It&#x27;s equal to zero and for our point to calm a zero. This is also there as well, since there are horizontal tangents. 12 a plus four B plus C is equal to zero then we&#x27;re going to subtract our equations. 12 day was for P. Thus he is equal to zero minus 12. A finest for P plus C is equal to zero. Then we got B is equal to zero. We&#x27;re gonna add up our equations. Our 1st 2 equations less for B plus two c rusty this equal to zero. Then we got Dee as equal to three. Since we plug in, zero for B C is equal to negative 12 A. When we get a was four times zero plus two negative 12 a plus three is equal to zero. Can we get a value of three over 16? And when we plug it in, C becomes negative nine over four. Then we just plug it in to get wise people to three over 16 x square x cubed Excuse me minus nine over four x plus three

Carson Merrill · Answer

So here we are given a curve with the equation Y squared equals three or x cubed plus three, expert. Okay, so with this in mind we want to implicitly differentiate um and find the equation of the tangent line at one negative two. So we see that when we calculate this we want to implicitly differentiate. So we&#x27;ll have to y Y prime equals three X squared plus six X. And then we evaluate this at X equals one. We see that the slope why prime is going to be a negative 9/4. So we end up getting Y equals a negative nine worth acts plus 1/4. And we see that that is tangent to the line one negative two as expected. Um And by graphing these on the same screen, we&#x27;re able to see that.

Paul Teng · Answer

focus for this problem. They tell you that there&#x27;s a quadratic function. Why equals a X worthless be ecstasy. And the Tolia has an X intercept of one. So we plug in X equals to one. Why should equals zero, which is this relationship we have right here. Zero able to able people see. And they also tell you it has ah y intercept of native to So whenever X equals zero whites and negative too. So technicals 20 a and B go away. So we end up with the relationship of may have to be able to see. And finally, they also tell you that the slope of the tangent line with the slope of negative one at the Y intercept, uh, is attention to the function. So we know if the tangent line has a slope of negative one at the Y intercept. So at the y intercept, um, the X equals zero right close to zero, I thought, Why intercepts? So the slope of the function at X equals zero. You goes to negative one, which is the slope of the tangent line. So knowing having all these information, we can go ahead. We&#x27;ve already got two systems right here in negative. You will see and zero equals Abel&#x27;s people. See, So I can go ahead and rewrite replace to see here with native to since we know that see, you call tonight too. And what we want to do is deal with this relationship, right? So if we take white prime, we&#x27;ll get to a ex post. Be So if we plug in zero for white time, we&#x27;ll get why prime of zero equals to be. We get a wide prime of zero. He goes to be with vehicles, too. Negative one. So now that we found R B and our see, all we have to do is find our A and using this relationship here we have zero able to a pause. People see. So we get zero. You go to a when this one minus two. So we end up with and a you goes three. So now we found all of our variables. You just plug them in so we get a quadratic function of three X squared minus X minus two, and that is your final answer.

Paul Teng · Answer

Okay, So for this problem they give you Ah, A general cubic function excuse will be X squared, but see exposed E And this problem has three parts A B and C. Uh, where part a. Do you want you to find points? They want you to find conditions for A, B, C and D so that the graph has to horse on pretensions to horizontal tenants for party one exactly one horizontal time for part B and zero horizontal titles for part C. So we&#x27;ll start off with part a with zero with two horizontal tangents. And the first thing we&#x27;re gonna need to do is to take the derivative of the function so we can find this slope, uh, slope with its corresponding tension line. So you think the powerful will get derivative off three X squared plus two b x plus C and D, the constant so that become zero. And to find the horizontal tensions, horizontal tended time, a slip of zero. Right. So we just have to set this derivative equal to zero and find the patterns that way. So for two horizontal tensions, that obviously means that the function will have two whores on the attendants and the only way that that will happen would be with a cubic function. Right. Since I caught a car, I function only has a maximum of one one horse uncle tendon and a line Odyssey does not have any horizontal time, so a cubic function we&#x27;ll have the possibility of having to. However, there is one exception. Words. There&#x27;s once an area where cubic function will have one horizontal tension. But we&#x27;ll get to that leader. So for a cubic function, the first thing we&#x27;re gonna need is that a cannot equal zero. They cannot equal to zero because if a equals zero, it will no longer be a cubic function. Now, for the next condition, we&#x27;re gonna have to solve for BNC. Now, how would we do that? Well, you can start by using the quadratic formula to solve for X, and this will help us differentiate the cubic function that has to horizontal tangents from the cubic function that only has one horizontal tension. So they&#x27;re using our hard on formula. They could be posterized word of B squared minus four. I see all over to a so bullion, our variables we get X equals in native to B plus or minus the square. Root off for B squared minus 12 a. C all over six A. And we can factor out off four from the square root to factor out of two and stuff will cancel and we&#x27;ll get end up with negative B plus or minus squared off B squared minus Tracy all over three a. All right, so now this would be this is solving for X And if you notice that if you notice will always have two solutions for X which will be our two horizontal tendons Unless unless our were almost this part of the quadratic formula right here, B squared minus three a. C. If that part equals 20 then will only end up with one solution and that solution would be negative be over 38 So it b squared minus three a. C equals zero. Then you end up with one solution. So one condition that we need is for B squared to not equal three a scene. So that is our second edition. And for D, since he is just shifting the graph up and down bow, Why access? It doesn&#x27;t really influence the amount of horizontal attendance we would have so Deacon be like and all real numbers do You can be all real numbers, so that would be for two horizontal tendons. These are the conditions for two or the attendants. No new. Create space for swollen from space for part B. Now hard be wants us to find petitions for A, B, C and D where there&#x27;s only one horizontal Qianjin one horizontal intention Let&#x27;s talk to him tangent All right, So if we&#x27;re dealing with one horizontal tension, we&#x27;re probably dealing with a quadratic functions and so those are parabolas on They have, uh, sort of attention either at their admit maximums or their minimums. Or as I said in the part A we&#x27;re dealing with you are cubic function that has one horizontal tagine at one point and that is ah, extreme years solution for here. So for our first our first one for our first scenario, our first set of conditions we&#x27;ll do will start with Cuba the Cuban function for a cubic function again, we need a to not equal to zero because then I would not want no longer be a cubic function. And the second about the red to zero on the second restraint is for when b squared minus three a C equals zero as a peace corps, my serious equals zero. Then we&#x27;ll end up with only one solution for X and the left prime. We&#x27;re at front of X now, if you only end up with one solution for X timings, we only have one or is all the attention. So if b squared equals 23 a c, we&#x27;ll get a cubic function with one horizontal tension and d again can equal anything. All real numbers since is only shifting the Grafton lot, influencing the number of horizontal tensions. Now, for our second set of solution, we&#x27;re going to be dealing with quadratic. So we want a two equals zero since a physical efficient for the X cube variable. And for our second stripper strain, we want be to not equal zero, since that is the cowfish in front of X word and we do want are quadratic functions. So with so that with those two restraints, C can be whatever it wants, because it won&#x27;t change the number of horizontal tango and same with D as always. So these are you&#x27;re too sets of conditions for one horizontal tensions a now swelling down for more space. We&#x27;re gonna finally do part C and Parsi is asking for zero horizontal Carrington&#x27;s Euro horizontal Kenyans on the only grass where there are zero horizontal tendons are for straight lines. And that only happens when a equals zero be also the hospital zero and then see And you can be whatever their long whatever they want, right, Because, uh, we want we&#x27;ll actually see cannot equal to see cannot equal to zero because of c was zero Then we end up with just We just had a You end up with just horizontal line itself And I guess horizontal line does have a horizontal tangent Technically so. So you cannot equal to zero and d can be whoever is want since actual sift shifting the graph up and down Yeah, why axis? So this is your final set for zero horizontal tensions

Bobby Barnes · Answer

we want to find the slope of the tangent to the curve. Why is it go to three plus four X squared minus to execute at point X is equal to a to start. Now the equation will need to use for this is going to be released one of the equations that I should say so f prime of a is equal to the limit as X approaches a oh Alexe minus half a day all over X minus a. Now let&#x27;s go ahead and plug everything in, which would give Oh so f of X was three plus four X squared minus two. Ex cute When we have f of A that just replaces all the exes with AIDS and then subtracting it should give us three minus or a squared plus two a cube all over X minus a No. Notice the threes cancel out. And the next thing I&#x27;m going to do is group all of the cube and square terms together and factor out the coefficients for the ex terms. So we&#x27;d have or X squared minus ace, where and then minus two times execute minus a cute all over X minus eight. Now notice that X squared minus a squared is the difference off squares. So that means we can divide X minus eight to it in the following way. So it should leave us with X plus a and then likewise over here. This is the difference of cubes and dividing that by ex mice. A should give us X squared plus a X plus a squared. Now, if we were to plug a in or evaluate the limit, I should say instead at a Well, this here is going to give us to a We&#x27;d get a bunch of a cubes right there. So be three a cube. So all we need to do is multiply it by four. And negative too. So that would give us a a minus six a squared. And this is going to be our slope of potential line at the point, eh? Now, to find our tangent lines, remember, we can use the point slope equation, which all rewrite as why is equal to M X minus X one plus by one. So at 0.1 by let&#x27;s go ahead and plug X is equal to one in tow. Find our slope so that would give us eight times one minus six times one squared, which should give us too. So we&#x27;re gonna have why is equal to two X minus one waas five and simplifying this down. We should get to X plus three. So this here is going to be our tangent line or this point here and for other point to three. So again, we need to find what f prime of two is, which is gonna be eight times to minus six times two squared. So that would give a 16 minus 20 or which is negative eight. So you have why is equal to negative eight X minus two plus three and that should give us negative. Let me write this Ballu Negative eight X Plus 19 or our tangent line or at this point here. Now they tell us to go ahead and Graff the standard lines along with our visual questions. I&#x27;ve already went ahead and did that, and you can see at our two points here it looks like they line up pretty well but the tangent lines So potential lights look like they only touch once, and also they just kind of go off as soon as they touch the function

Problem

Find a parabola with equation $y=a x^{2}+b x+c$ t…

Problem 65 Hard Difficulty

Find a cubic function $y=a x^{3}+b x^{2}+c x+d$ whose graph has horizontal tangents at the points $(-2,6)$ and $(2,0)$ .

Answer

$y=\frac{3}{16} x^{3}-\frac{9}{4} x+3$

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$y=\frac{3}{16} x^{3}-\frac{9}{4} x+3$

Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus Early Transcendentals

Grace H.

Anna Marie V.

Kristen K.

Samuel H.

James Stewart

Calculus Early Transcendentals