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Sketch the graph and show all local extrema and inflection points 12) y-x4 4x3...

Question

Sketch the graph and show all local extrema and inflection points 12) y-x4 4x3

Sketch the graph and show all local extrema and inflection points 12) y-x4 4x3



Answers

Graph the polynomial, and determine how many local maxima and minima it has. $$y=x^{3}+12 x$$

Okay, So to find out the special values for dysfunction first so we can do is we take that first derivative so f from vets that's going to be equal to six times 2 12 extra moneys. 12 extra third. Nothing the equal to zero we can send equal is there's we could find these critical values. I went there, and I see here that we can pull out is a 12 x to third. Just gonna leave us with, uh, X squared minus one song is equal to zero. So clearly it told Exit third is equal to zero. That means X plus B equals zero. Then X squared minus one equals zero. We would want other side sex. Critic was one expert secret of positive and negative one. So that street critical values there that we found just put a number line up right quick zero negative one and one. And it if we were to pick a number less than negative one between negative infinity negative on and put it into F of X, we get a largely negative number showing that for FX in that integral is decreasing. And then if we chose a number between negative on a zero uploaded to F problem of eggs were going to get a positive number showing that in that interval is increasing, then between zero and one that we will get a negative number for plugging my number in between that for f prime of X, so decreasing again and then anything greater than when we're going to get a positive number for convicts. So then defensive eat, increasing so clearly big own because of this craft, Would you see that negative one of one our minimum values while zero is an axle. Now let's look at the second review a double prime of X. This will show us our inflection points. Okay, so we're gonna most by five times told we're going to get 60 x to the fore minus 36 x squared were set that equal to zero to find the infection quakes. Okay, I can carry out a 12 x squared out of that and we'll be left with five x squared. Minus three said they could zero. So 12 X squared is equal to zero. Excellent Speaker zero and then five X squared. Mostly quick. Three X squared would be good if 3/5 accidentally Good applause of *** scared of Greek over five So that's on 23 inflection points right there. Now let's look it to see if we said every set X equaliser So go zero is going todo was your my 00 So clearly 00 is gonna be We said What? X equals zero system your white Why interested one x and y interested And then the sea. When y is equal to zero, we're gonna have to x to six my street x fourth and so something else you could take out is ah X squared. No, sorry. Exit before. So for ex support equal to zero, clues are zero. Must be the answered wearily Fonda in the previous step. So we're just gonna cancel that out, So I'm going to disappear. Don't say two X squared minus. Wasn't he could three when you moved at, uh, three over to the side. So x squares when we goto 3/2 X is going to put a positive negative screw 3/2. And that's gonna be for while Isaac Reserve. So let's list everything we've got. So we have 00 That is, uh, a local max. It's also our Brazilian X and Y intercept, and then is also named collection. Then let's look at when you put in negative one and when you put it in one so X equals negative one. Uh, fx also gonna be good negatively when x is equal and one is also going to be able to negative one. And these are actually gonna be absolute minimums for function. And then we have square with the revive Enigma Square. Do you have I? And when we put that in, we should see the for both of these. They're gonna be equal to negative six for eight. And these are gonna be inflection points on efforts, girl 3/2 and negative script 3/2. Y was equal to zero. So these are ex intercepts. So why don't we put all these into a graph? Just a rough sketch, Right? We have. Ah, say Syria. Zero things are going. Thank you, boy. One negative one. Okay, we're gonna have positive native script. You find negative 6.8. So to me, be somewhere around here, and then we're also gonna have positively sphere screen would be over 20 So I wonder you're on, like here. So? So the graph coming is gonna look something like this. Why? Might look a little weird. It's just that the graph flattens out here between positive and negative script. You were to those values and they're very close together. Your home 20 But that's what the graph should look like.

Hello. So here we have. The function F of X is equal to X cubed minus 12 X. So this function is gonna be defined for all points. So therefore the domain of the function is negative infinity. To infinity. Now find the critical points we differentiate here with respect to X. And find that the first derivative F. Prime of X. That's just equal to three um Times acts to the three minus one. So that's a three X squared and then minus 12. This is the first derivative. Um And then we um find the points where our first derivative is equal to zero. So we take our first derivatives set it equal to zero. And we see that if three X squared minus $20.0 that implies that three X squared is equal to 12. Which implies that X squared is equal to four inscrutable sides implies that X is equal to positive or negative too. So the derivative here is defined for all points. Therefore the critical points are X. is equal to negative two X is equal to two. Then the second derivative. What we differentiate our first derivative. So the derivative of three X squared minus 12. That's going to be six X. So the second derivative f double prime of X is going to be equal to six X. Then we do the second derivative test. So we test um from while negative 2 to 16. Um and the sign of the second derivative um at at the point negative to um it's gonna be less than zero. So we have a relative maximum and then um After a point of two is Native 16. Um So therefore we have a relative minimum. Um So the um we can then go ahead and we can um grab this thing as we see here. So here we see um the graph of again f of X equals X cubed minus 12 X. And we see that we have right here. Um The point negative to 16, right? That's going to be a relative maximum. And then we have while we change con cavity, right at an inflection point here, at the origin, at the 0.0.0 and then we have a local minimum here at the 0.2 comma native 16.

Okay, So find it extremely inflection points and extend my innocents, but its function First, we're going to take the first derivative f of X, so that part of it is gonna be equal to three punished three x square, and then we're gonna set that equal to zero. That means that we x squared with the equal three x squared, equal while or accident equal a positive or negative one. So let's put down on a number line or quick negative oil causes abroad. And when X is going to be less than negative one f prime of X is going to be negative, meaning that the function is gonna be on going downwards up until a negative one and then in between negative one and causes of one at problem of X is gonna be positive. Meaning that it's gonna be sloping upwards is gonna be going up roots or increasing between that interval until you get so one where anything greater than one is again. Now you do. This is gonna be all decreasing. The function can be decreasing anything greater than what So we could tell here that negative one is going to be a minimal value possible and gonna be a maximum value. Right. And to find our inflection point, you have to take the second bridges. I have a double problem X, which is going to be equal to negative six x squared, and we're gonna start *** six X. I was gonna be equal to zero. So clearly, Excellency, equal to zero. That's gonna be a reflection point as well as our X and y intercept. Because when X is equal zero, why is equal to zero? So we have negative ones. And when X equals negative one, why is equal to negative two? That is gonna be our local minimal. And then we have one. When X is equal to while why should be equal to two that's going to be equal to Arlo. Cool, Relax. And then 00 is gonna be our inflection point. And our X and Y intercept is well

Okay, So find it extremely inflection points and extend my innocents, but its function First, we're going to take the first derivative f of X, so that part of it is gonna be equal to three punished three x square, and then we're gonna set that equal to zero. That means that we x squared with the equal three x squared, equal while or accident equal a positive or negative one. So let's put down on a number line or quick negative oil causes abroad. And when X is going to be less than negative one f prime of X is going to be negative, meaning that the function is gonna be on going downwards up until a negative one and then in between negative one and causes of one at problem of X is gonna be positive. Meaning that it's gonna be sloping upwards is gonna be going up roots or increasing between that interval until you get so one where anything greater than one is again. Now you do. This is gonna be all decreasing. The function can be decreasing anything greater than what So we could tell here that negative one is going to be a minimal value possible and gonna be a maximum value. Right. And to find our inflection point, you have to take the second bridges. I have a double problem X, which is going to be equal to negative six x squared, and we're gonna start *** six X. I was gonna be equal to zero. So clearly, Excellency, equal to zero. That's gonna be a reflection point as well as our X and y intercept. Because when X is equal zero, why is equal to zero? So we have negative ones. And when X equals negative one, why is equal to negative two? That is gonna be our local minimal. And then we have one. When X is equal to while why should be equal to two that's going to be equal to Arlo. Cool, Relax. And then 00 is gonna be our inflection point. And our X and Y intercept is well


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