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[0/0.13 Points]DETAILSPREVIOUS ANSWERSSCALCET8 3.1.055Find the points on the curve Y =213 3x2 12Xwhere the tangent line Is horizontal:(x, Y)(smaller X-value)XY)(lar...

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[0/0.13 Points]DETAILSPREVIOUS ANSWERSSCALCET8 3.1.055Find the points on the curve Y =213 3x2 12Xwhere the tangent line Is horizontal:(x, Y)(smaller X-value)XY)(larger X-value)Enhanced FeedbackPlease try again Find the x-value whereequa to 0 This is where the tangent of the grapNeed Help?Raad ItWatchItTdkto IterSubmit Answer

[0/0.13 Points] DETAILS PREVIOUS ANSWERS SCALCET8 3.1.055 Find the points on the curve Y =213 3x2 12X where the tangent line Is horizontal: (x, Y) (smaller X-value) XY) (larger X-value) Enhanced Feedback Please try again Find the x-value where equa to 0 This is where the tangent of the grap Need Help? Raad It WatchIt Tdkto Iter Submit Answer



Answers

Find the slopes of the tangent lines to the curve $y=x^{3}-3 x$ at the points where $x=-2,-1,0,1,2$.

Hello. I'm going to be working with this curve y equals two x cubed plus three X squared, minus 12 X plus one. And we're going to be looking for At which points is the tangent of this line going to be equal to zero? So we're going to start off by going over to Dismas and looking at this curve. I already put the equation for the curve into Dismas. So here is the graph of this line. So we would expect the tangent to be horizontal at the maximum and minimum on. This is because the tangent is the slope and the slope starts steep down here as it gets closer to the maximum. It kind of levels out before tipping in the other direction to start going down, and then at the minimum, it levels out to zero and then tips back upward. So we're going to expect the tension. To be zero at these two points will keep that in mind while we're going through this problem. So we're going to start off by finding the first derivative of this equation on. That's because the first derivative of a slope of a line is the slope of that line. So we're going to do why Prime equals for the first term. We're going to multiply the constant of two by the exponents of three. So we're going to get six X and then the exponents of three subtracted by one is going to be too. So that term is going to be six x squared. Similarly, for the next 13 times two is six x two minus one is one minus. Um, there's, ah, an implied one here. So 12 negative, 12 times one is negative, 12 x one minus 10 And then, since there's an implied X to the zero over here, one time zero is gonna be zero. So that constants going to cancel out. So the tangent is going to be horizontal when the slope of the line is equal to zero. So using that, we're going to set this the derivative equal to zero, and we're just going to use this same equation minus 12th. We're going to simplify that by factoring out of six because we're trying to, um, we're trying to isolate those excess, and we want it to be a simple as possible zero divided by six is still zero, so we have zero equals X squared plus X minus two. So this equation is easily factory herbal because we know that the factors of negative to our positive and negative one and two So we need to find factors that are going to multiply to be negative, too, and add to be positive one for the middle term. So we see that if zero equals X minus, one times X plus two negative one times positive two equals negative two and negative one plus two is equal to positive one, So this is going to factor out properly. So now that we have this either one of those terms, either X minus one or X plus two is going to need to be equal to zero or both. Of those terms need to be equal to zero for the slope to equal zero. So we're going to take each of those. In turn, zero equals X minus one. Adding one to both sides. We see one equals X, so there's our first X point. Second, we have zero equals X plus two and then subtracting two from both sides. We get negative. Two equals X so there's our second X point so we can see that X is going to be equal Teoh one or negative too. So now that we have the X points, we need to find the y ports to kind of to see at which points those tangents are going to be horizontal for so to find the white points were actually going to go back to the first equation without the derivatives. So we no longer need the derivatives. I'm going to delete that, and I accidentally deleted the why as well. So here we go. So we have why equals two x cubed plus three X squared minus 12 x plus one. That's the original problem. So to find the y co ordinates, we're just going to plug in each of those excess in turn so we'll start out with X equals one. So why equals two times one cubed plus three times one squared minus 12 times one plus one. It's going to be a Y equals two plus three minus 12 plus one, which equals negative six. So our first point is going to be one negative six. Next, we're going to use X equals negative too, and we see why equals two times negative to Cube plus three times negative two squared minus 12 times negative two plus what you know, simplify that more two times and negative. Eight plus three times four minus 12 times negative two plus one. And why equals negative? 16 plus 12 plus 24 plus, but one is equal to 21. So our second point is going to be negative two and positive 21. So let's go back over to Dismas. So we originally assumed that the high point and the low point We're going to be thes allusions to this problem. So we're going to use one negative six and we see that the minimum is in fact, at one regular six that we found and negative to positive. 21 is brat maximum, which is we also expected to have the horizontal tangent line. So that's how you find the horizontal tangent line of this problem

Hello. We're going to be working with the curve. Why equals two X cubed plus three X squared minus 12 x plus one we're trying to find at which points the tangent of this line is going to be horizontal. So to start off, we're going to need to find the derivative of this life because the derivative is the slope of a curve. So we're finding why prime equals. So for the first term, we'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're going Teoh look for where the slope is going to be horizontal, in other words, where it's equal to zero. So why prime that slow Klein equals zero equals, and it's just going to be that first derivative equation. We're setting that 20 so we want to simplify that by factoring out a six because we'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'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'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'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'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's going to be our first coordinate for this horizontal tangent line. Next, we're going to see X equals negative, too. I'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'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'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.

Hey, guys. So you'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

Hey, it's Claire is the one you married here, So we're first gonna differentiate her function. We got six x square plus six X minus 12. We're gonna study equal to zero to find where the points where the tension line is horizontal. You get six x plus two times X minus one. So we see that X bias are negative, too. And one, we plug this in to our original equation to negative two comma 21 and one common negative six.


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