find-a-cubic-function-y-ax3-bx2-cx-d-whose-graph-has-horizontal-tangent-at-the-points-2-6-and-2-0

Name: find a cubic function y ax3 bx2 cx d whose graph has horizontal tangent at the points 2 6 and 2 0
Uploaded: 2017-10-31
Duration: 4 min 16 s
Channel: Numerade
Description: Find a cubic function $ y = ax^3 + bx^2 + cx + d $ whose graph has horizontal tangent at the points (-2, 6) and (2, 0).

Question

Find a cubic function $ y = ax^3 + bx^2 + cx + d $ whose graph has horizontal tangent at the points (-2, 6) and (2, 0).

Wesley Hines · 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.

Shumayal N · Answer

So first we&#x27;re gonna find the equation for the tangent at any point on the line, and that is the derivative of the function that&#x27;s been given to us. And we see that there is a term in the numerator and in the denominator. So we&#x27;re gonna need to use the kosher soul in order to get the derivative according to the quotient rule. First you use you leave the denominator as it is in the numerator. So execute minus six x minus one stays at as it is, and you take the derivative off the numerator and we&#x27;re going to do that using chain rule. So first, take the derivative of the outer function, which is the power part of the function for two into X squared minus one, and then take the derivative of the inter function X squared minus one, which is two X and then Linus. Now you leave the numerator as it is. So X squared minus one squared stays as it is, and you take the derivative of the denominator, which is three X squared, minus six. And this entire thing is divided by the denominator squared, so X cubed minus six. Next minus one squared. Now, in order to get me value for the Grady int at the points that are given, you substitute the value of accents, the equation. So when you substitute zero into the equation, you will get the gray didn&#x27;t as equal to six. Now the equation of the line is y equals mx plus e. So we need to get the y intercept, which is C So we just substitute me. Why the X and the greedy int into the equation and from this r C equals minus one. So your final equation of line at a given point is why is equal to six X minus one.

Will Erickson · Answer

either. In this problem, we are asked to find the equation of a tangent line, Uh, at the given point for this curve. So the curve given is X Y plus X squared y squared equals five and her point will be two comma one. So, as always, when we want the equation of a tangent line, we have the point. We&#x27;re gonna end up using point slope form. The point is done. We just need the slope, which means we will need the derivative. So let&#x27;s begin. We can see right away we&#x27;re going to need implicit differentiation since the wise are all mixed in with the exes. So let&#x27;s take the derivative of the left and of the right and see what we get unless we&#x27;ll need the product rule for both of these, since we have X times why they&#x27;re so that&#x27;s used. The product room, the derivative of X y is X times the derivative of y, which is why prime or one times why prime plus the derivative of X is just one. So the product rule says derivative is X squared times the derivative of y squared, which is to why, But then we need to multiply by. Why Prime? That&#x27;s half the product rule. And then plus the derivative of X squared is to x on times the second part, which is just what I squared. So we did there. We used product rule the first term and the product fool on the second term. Okay. And then for the right side, the derivative of five is just zero. So at this point, all we have to do is get the white prime isolated here. So let&#x27;s look to see we have that term has a y prime, and so does this third term. And so we will distract. That&#x27;s a tract all the terms that don&#x27;t have a y prime. So we will subtract this term, Which is why who also subtract this term should minus two x y square. We&#x27;ll do that from both sides, which gives us X y prime plus C two x squared y y prime on the left and on the rights. We just get those two terms be subtracted from the left. Okay, Now we are ready to factor out that why prime from the left? And once we do that, we are basically, uh, one step away. So why Prime ends up being once we divide by everything on the left this&#x27;ll fraction here. That&#x27;s why prime now all that&#x27;s left to d&#x27;oh is plug in to for X and one for Why, since we want the slope at that point. So really, what we&#x27;re interested in since our slope. But this tangent line is going to be the derivative specifically at the 0.2 comma one and let&#x27;s just see what happens that&#x27;s plugged into for X and one for why? So we get minus. Why is one just two times X is too. Why is one then on bottom access to exes to And why is one all right? So on top, you get negative one, my ass four And on bottom we get two. Plus it should be ate, it looks like which is negative five on top 10 on bottom. That reduces to negative 1/2 That&#x27;s our slope. And now that we have the slope, we can answer the question. So the equation of our tangent line Remember the point slope form? Why, minus the point of the value of our point we know is one equals slope. It is negative half was. Put that into the slope Times X minus. The X coordinate we know is too. So there&#x27;s the final answer. If you want to put it into slope intercept form, you certainly can. But the problem doesn&#x27;t specify you have to. So there&#x27;s our Tanja line that out.

Kira Schwander · Answer

Hello. We&#x27;re going to be working with the curve. Why equals two X cubed plus three X squared minus 12 x plus one we&#x27;re trying to find at which points the tangent of this line is going to be horizontal. So to start off, we&#x27;re going to need to find the derivative of this life because the derivative is the slope of a curve. So we&#x27;re finding why prime equals. So for the first term, we&#x27;re going to use the constant too times the exponents three get six X and then three lowered by one is too, plus the constant three times the exponents two is six again X and then lowering the exponents, we get an exponents of one next, the constant of negative 12 times The exponents of positive one is negative 12 x lower the exponents by 10 plus. And then for that final term, we have ah, an implied X to the zero because any number to the zero power is going to be one. So we have the constant one times the exponents zero is equal to zero. So that constant just kind of cancels out at the end. So now we&#x27;re going Teoh look for where the slope is going to be horizontal, in other words, where it&#x27;s equal to zero. So why prime that slow Klein equals zero equals, and it&#x27;s just going to be that first derivative equation. We&#x27;re setting that 20 so we want to simplify that by factoring out a six because we&#x27;re trying to isolate the X as much as possible to find which X coordinates correspond with these horizontal tangent lines. Zero divided by six is still zero. So dividing both sides by six get zero equals X squared plus X minus two. Now this line equation is easily factory herbal because negative to has only the factors positive and negative one and two. So we need to find factors that will multiply to be negative two, while adding up to be positive one for that middle term. So if we have zero equals X minus, one times X plus two negative one and positive to multiply, to be negative, too, and they add to be positive one. So this is going to be our factors. So for this equation now, since it&#x27;s set to equal to zero either one of the terms, X minus one or X plus two needs to be equal to zero for these to multiply to be zero. So we&#x27;re going to look at each of these in turn, so zero equals X minus. One is going to be true if we add one to both sides and see that X is equal to one. Likewise, if we have zero equals X Plus two, we can subtract two from both sides and have negative two equals X. So our two X points are going to be X equals one and negative, too. And I&#x27;m going to write those up at the top so that we can reference those later. So now that we have the X points, we need to find the y points so that we can find the exact coordinates at which the tangent line is going to be equal to zero. So for the to find the Y coordinates were just going to go back to the original line equation and plug those excess in. So we no longer need the derivative equation. So I&#x27;m going to clear this up a bit. So going back to the original equation we have, we can see by plugging in X equals one. Why equals two times one cubed plus three times one squared minus 12 times one plus one. Simplify that to get why equals two plus three minus 12 plus one, which equals negative six. So when X equals one y equals negative six. So that&#x27;s going to be our first coordinate for this horizontal tangent line. Next, we&#x27;re going to see X equals negative, too. I&#x27;ll see why equals two times negative. Two cubed plus three times negative. Two squared minus 12 times negative two plus one Go simplify that two times negative eight plus three times for minus 12 times negative. Two plus one will simplify that one last time to get negative 16 plus 12 plus 24 plus one, which equals positive 21. So when X equals negative two y equals 21. So that&#x27;s going to be our second coordinate. And if you look at the original line, um, graph on a graph, you can see that the coordinates one negative six and negative to 21 are the maximum and the minimum of this line, and that is where we would expect to have a horizontal tangent line. So that&#x27;s a way that you can go back and check to see whether all the math worked out the way that you would expect it to thank you.

Caleb Miller · Answer

Hey, guys. So you&#x27;re asked to observe this equation and find where the tangent is horizontal Now, horizontal tangent is the same thing is asking where is the derivative equal to zero? So do that. First, we will derive this equation. When we do that, we get why Prime of X is equal to six X squared LA six x minus 12 Now even said that equals zero solved. We it six x squared. Ah, six seconds minus 12. Secret zero to simplify, weaken, divide through on both sides by six but by six on this side and on this side, When we divide out, we had X squared plus X minus two equals zero. This factor is pretty easily we get X plus two X minus one is equal to zero. We could have also used the quadratic formula to find those roots. Now we can write our answer. We have X seals negative too and x table two positive one. At those points, the slope will be equal to zero and thus the tangent will be horizontal. Thanks for watching

Frank Lin · 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

Problem

Find a parabola with equation $ y = ax^2 + bx + c…

Problem 69 Hard Difficulty

Find a cubic function $ y = ax^3 + bx^2 + cx + d $ whose graph has horizontal tangent 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

Catherine R.

Kayleah T.

Michael J.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Catherine R.

Kayleah T.

Michael J.

Joseph L.

James Stewart

Calculus