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[) For f(x) - * -* 3x + find the following and then sketch the graph. DomainFirst DerivativeIncreasing Intervals Decreasing Intervals Relative Max Relative Min_ Sec...

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[) For f(x) - * -* 3x + find the following and then sketch the graph. DomainFirst DerivativeIncreasing Intervals Decreasing Intervals Relative Max Relative Min_ Second Derivative Concave Down Intervals Concave Up Intervals _ Points of Inflection _2) For cos X - 2sin X, 0 sX s 21 find the following and then sketch the graph DomainFirst DerivativeIncreasing Intervals Decreasing IntervalsRelative Max

[) For f(x) - * -* 3x + find the following and then sketch the graph. Domain First Derivative Increasing Intervals Decreasing Intervals Relative Max Relative Min_ Second Derivative Concave Down Intervals Concave Up Intervals _ Points of Inflection _ 2) For cos X - 2sin X, 0 sX s 21 find the following and then sketch the graph Domain First Derivative Increasing Intervals Decreasing Intervals Relative Max



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Sketch the graph of each function. Indicate where each function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down, where any points of inflection occur, and where any intercepts occur. $$f(x)=\frac{x+1}{x^{2}-2 x-3}$$

Uh huh. We want to draft Halfbacks equals one over X squared plus three. First. We analyzed after basic info on the graph. So the domain is all X. There's no Y intercept the horizontal axis is like zero and there's no vertical. As soon as the denominator cannot equal zero. Next we analyze F. Prime for where the function is increasing or decreasing. F. Prime is negative. Two X over X squared plus three squared. The petitioner critical point is X equals zero, so to the left of X equals zero. Prime is positive. Is increasing to the right is negative. F is decreasing. Next we analyzed edible products, cavity purposes. So ethical prime is six times square minus one over expert was three cubed. The petitioner critical point where Mexico sponsored minus one. It's less of negative one. Ethical practice positive is increasing between negative one and one X times negative. Uh It's gonna keep down and to the right uh one at a time as positive fever. Therefore we have the graph of the afternoon right. We have to change the cavity at plus or minus one noted, as well as a change from increasing to decreasing at X equals zero noted.

Okay, so we need to find the extreme of the points of inflection where the function is con cave up where it is concave down where it is increasing and where it is decreasing. So we have the function negative X cubed plus three X plus two. So we can find the extreme where the first derivative is equal to zero. So the first derivative is negative three x squared plus three. So this gives us X is equal to plus or minus one So exit negative one. Our function is at zero an ex at positive one or function is at four. So these are our extreme Now we need to find where we have inflection points. This is done where the second derivative is equal to zero So the second derivative is negative six x So this gives us that X equals zero when x equals zero r function is equal to two So zero comma to is our inflection point. Okay, so now let's look at the first derivative to find where the function is increasing or decreasing. It is increasing where the first derivative is greater than zero. So we can see from negative infinity to negative one. The function will be decreasing from negative one toe one, the function will be increasing and from one to infinity the function will be decreasing. And now we look at the second derivative to see where the function is concave up or concave down where the second derivative is greater than zero. The function is con cape up so we can see from negative infinity to zero. The function is con cave up and from zero to infinity the function is con cave down. This is the sketch of negative X cubed plus three X squared plus two Using the results that we just found, we have a minimum at negative 10 a maximum at 14 and an inflection point at 02 Our function is decreasing from negative infinity to negative one. It is increasing from negative one toe one and it is decreasing from one to infinity. The function is a con cave up from negative infinity to zero and it is con cave down from zero to infinity. So this is an appropriate sketch of our function

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.

Okay. We need to find the extreme of the points of inflection where the function is increasing or decreasing and where it is concave up or con cave down. We have the function, the quantity X plus one to the two thirds power. Okay, So first we need to find where the first derivative is equal to zero or it does not exist. So we can see that the first derivative is equal to to over three X plus one to the two thirds power. We can see that you haven't asked him. Tote in this function and your ass in Toad is at X equals negative one. So this is one of our extreme. Um It has the coordinate negative one comma zero on our graph where the second derivative is equal to zero or does not exist. The function has inflection points. So the second derivative is equal to negative 2/9 X plus one to the negative 4/3. We can see that dysfunction does not ever cross zero. It is always less than zero. So there are no inflection points where the first derivative is greater than zero. The function is increasing. So we can see that the function is decreasing from negative infinity to negative one. Then it is increasing from negative one to infinity. And as we just said, the second derivative is always less than zero. So this tells us that our function is always con cave down. So it is con cave down from negative infinity to negative one. There's a break Then again from negative one to infinity. So this is what our sketch will look like Based on our work, we have a minimum negative one common zero. We have no inflection points. It is always con cave down and it is decreasing from negative infinity to negative one and increasing from negative one to infinity.


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