Here we have the function and I'm going to begin with the horizontal assume totes um The degree of the denominator is greater than the degree of the numerator and so horizontal a sento will be at zero. Also f of zero is three over zero plus one which is three interesting. Um Okay I'm going to take the derivative of the function and in order to do that I'm going to rewrite the function. So I'm gonna move this over. Whoa. Not that far. Okay. I'm going to rewrite this function as three X squared plus one to the negative one power. So they can use the chain role. So using the chain role negative three X squared plus one to the negative to power times the derivative of the inside which is two X. Yeah. Which I can simplify to negative six X X squared plus one to the negative to power. All right, max and min. Well this occurs when the function equals zero and the numerator would have to be equal zero. So negative six X. Would have to be zero and therefore X equals zero. So there would be a max or min had X equals zero. So I need to be concerned about the integral from interval from negative infinity to zero and then from zero to infinity. So I'm going to substitute in one. Um and it's going to give me negative six times one times one squared is one. Okay. That is clearly a negative number. And so the slope is negative in this interval and it's positive in this interval. Okay, let's move on to the second derivative. I'm gonna have to use the chain role and the product rule for this. So uh derivative. The first is negative six times the second. Okay. Yeah. Plus derivative of the second which is negative two derivative of the inside times the first. Okay, now I got to simplify this. Um um um I'm, mm Okay well at the very least I can factor out a negative six and I'm going to write this as one over X squared plus one minus. Uh huh. Forex times X squared plus one. Yeah. Okay. Well that didn't accomplish whole lot there. But wait a minute. I think I made a mistake here because um the derivative of X squared plus one to the negative to power would be negative two X squared plus one to the negative three power. Then the derivative of the inside then that so not a big deal other than I mean this is pretty quick one to fix. Um Mhm. Okay. X squared plus one. Mhm. Cameron cubed. Okay good. Yeah so let's try simplifying this. Okay X squared plus one squared X squared plus one cube. So if I multiply that first term Buy X squared plus one squared in the numerator and the denominator I get this then I have the same denominator so I can subtract four X. And maybe that's a little bit better. Um X squared plus one square. That's going to be X. To the fourth power plus two X squared plus one minus four X. Man. I hate it when that happens X squared plus one. And by that I mean nothing canceled out. Okay. Inflection point. Yeah. Well this will occur when the numerator equal zero. That's supposed to be a four here. I thought this would be easier than the previous one but it's not extra fourth power plus two X squared minus four X plus one equals zero. Mhm. Okay I'm gonna put this into a graphing calculator and uh see where that equals zero X. To the fourth. That's two X. Square and it's four X. Plus one. All right, That equals zero at at least two places. And I'm zooming out to see if there's any others. But now it looks like just two places. So let's get those spots. Yeah 0.296 and one one. So those are the inflection points. Okay so now I've got to think about intervals from negative infinity 20 point 296 Then from zero point 296 to 1. And then from one to infinity of course the easiest thing to check. Zero. So let's put that into the equation um negative six. I put in zero for X. Almost everything cancels out. And I'm just left with one in the denominator. I'm going to have zero squared plus one. It's just going to be one. And so it is negative negative inflection. That would make this one positive inflection and that would make this positive inflection. Okay, now, I don't think I figured out the value at the max And min, although there's there's only one max, it's a zero. So F of zero is three and that is the maximum. All right. So I can go ahead and start to graph this. The 0.0 comma three is the maximum. And coming from there, the slope is positive leading up to it and negative after it. And it also has an ascent owed at zero. Ask them to go to zero assassin 2 to 0. Now my inflection points don't seem right because this is symmetric. So I think that I've made a mistake on my inflection points. And so now I gotta go back through and try to figure out what went wrong. So, here's how I'm going to decide to do this. I have F prime of X and I'm confident that that's correct. I just I just check some stuff negative six X over X squared plus one squared. I think that's what it said. Let's go back up again, negative six X over X squared plus one squared. Last time I used the product role. This time I'll use the quotient role and see if I get the same answer. So um denominator I'm going to write F double prime of X. It's going to be the denominator times the derivative of the numerator times the derivative of the new numerator, which is negative six minus the numerator times the derivative of the denominator, derivative of the denominator is too now the derivative of the inside. Okay. Over denominator squared. This seems very similar to what I did last time. Yeah. And it should be because they should both work. Now I see that I've got an X plus one X squared plus one. Lots of places. That should have been a four down there. Okay. Because I had to do denominator squared. Yeah. Cross that out right of three. Okay, so now I've got negative six X squared minus six plus. Um Six times two is 12 times two is 24. This is very different from what I had last time. So I clearly made a mistake last time. Um 24 X squared ok. 24 X squared minus six X squared is 18 X squared. Okay. Inflection point where the numerator equal zero adding six to both sides and then dividing by 18. Mhm. This makes more sense simplifying that. Okay, that makes more sense. So there was a mistake in all of this, this derivative up here. I'm not gonna search for it. But what I did was I switched to using the quotient role. That of course, does not explain the mistake at all. So, negative infinity to negative squared of 3/3. Negative square to 3/3. Two squared of 3/3 square root of 3/3 to infinity. And I'm going to check zero because it's in this range here. So where's my simplified thing here? Okay, good. If I put in zero, I will get negative six over one. So, clearly this is going to be negative, positive, positive. And that makes sense here. That can cavity is negative and then here it's positive and here it's positive. All right. Thank you for watching. So, I think what I'm demonstrating here is if you happen to make a mistake, then you can go back and try it a different way or which I I didn't do it this time. You could look at what you did and and find the mistake. But I just decided to do it a different way. All right. Thank you for watching. Uh huh.