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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 $(2,0)$ .

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

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Differentiation

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

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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'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'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's equal to zero. If we announce your attention towards the 1st 2 equations, we'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'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'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'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'll get eight a plus two times. See? Well, as we said, See is you got a negative 12 a. So I'm going to substitute that out and then let'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's equal to zero. And then from here, solving out we'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'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.

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