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Consider tlee Funetion f()State the domain of f(x) anc write the eqjuatious for the vertical nsymptotes of f(r). (6) Exaluate tlue following:f()lim f(r)lim f(r) =li...

Question

Consider tlee Funetion f()State the domain of f(x) anc write the eqjuatious for the vertical nsymptotes of f(r). (6) Exaluate tlue following:f()lim f(r)lim f(r) =lim f ()(c) Find the intervaly of inerevtst' And dxrensc. (d) Fiud the ($,W) coordinates of the local Iaxitunus #nd MIIAS Fiud the intervals ot cxncavity Find the inflection pints possible. If not jursibk, write DNE: 81 Evalunte thc following;liw f(r)f(t) =(b) Write the cquation for the horizontal asvmptote of f(c). Use Yolr JUISWC

Consider tlee Funetion f() State the domain of f(x) anc write the eqjuatious for the vertical nsymptotes of f(r). (6) Exaluate tlue following: f() lim f(r) lim f(r) = lim f () (c) Find the intervaly of inerevtst' And dxrensc. (d) Fiud the ($,W) coordinates of the local Iaxitunus #nd MIIAS Fiud the intervals ot cxncavity Find the inflection pints possible. If not jursibk, write DNE: 81 Evalunte thc following; liw f(r) f(t) = (b) Write the cquation for the horizontal asvmptote of f(c). Use Yolr JUISWCES MNJa L sketch thee grnph o flr) on the nxis below .



Answers

(a) Find the vertical and horizontal asymptotes.
(b) Find the intervals of increase or decrease.
(c) Find the local maximum and minimum values.
(d) Find the intervals of concavity and the inflection points.
(e) Use the information from parts $ (a) - (d) $ to sketch the graph of $ f $.

$ f(x) = x - \frac{1}{6}x^2 - \frac{2}{3} \ln x $

For this question with in party we look for where? Everything for the most sympathetic behavior. So full of verticals and Celtic ways. See that the domain off if is from zero to infinity. So we look for this limit. X coast is here from the right and we find out this actually caused to infinity. So there is a There's a vertical sympathetic at, uh a little goes a Metallica XY close to zero in the for the horizontal sympathetic revalue, its limit exposed to infinity and the extra cost of minus infinity. So there's no no heretic, no horizontal sympathetic in this case for part e for the increasing and decreasing tomorrow we right out to the first of the curative, which is minus X minus one x minus two, divided by three x. So if prying, because a zero gives us xy costa one or etc cause or two. So we have the reserve intervals from 0 to 1 no 1 to 2 in the front or to infinity on the first interval. If prime is negative, so the function is decreased on the second place. If prime is positive, the functions increasing on the last interval if primaries negative again. The functions de quizzing. So that means Naser there is a local I mean het xy close the ones. So if one, um, you cost over 5/6 in the less a local minimum on local next month at X equals 22 the value will be left to close to four with reminders to third normal too. For puffy right out The second of narrative which is two minus X squared, divided by three x square. So if the prime because of zero we have X equals two class minus two. But yes, minus put off to is outside of the men so we don't need to consider this point. So we have to stop intervals from the old to write off two in the front route off to infinity on the first interval Double primaries, Politics of the fortunes country of our who was thinking about the experience Negative. So the fund your contract down And then there's the inflection points. That exit was the route off to Now we are really the graph. So the function pony Define bones zero to infinity. Um so firstly come is decreasing can keep up. They hit the, um, reflection point at X equals the rooftop to So you hit the, um X equals toe one, which is a local Milliman noted inflection points. So then the cup that no function is increasing, But when you increase, you change it. Contiguity is here, so there is a reflection point and, uh, he hit another local extremely, which is a welcome 80 minute X equals two the staff people issue. So this is a sketch of the graph.

For this question with in party we look for where? Everything for the most sympathetic behavior. So full of verticals and Celtic ways. See that the domain off if is from zero to infinity. So we look for this limit. X coast is here from the right and we find out this actually caused to infinity. So there is a There's a vertical sympathetic at, uh a little goes a Metallica XY close to zero in the for the horizontal sympathetic revalue, its limit exposed to infinity and the extra cost of minus infinity. So there's no no heretic, no horizontal sympathetic in this case for part e for the increasing and decreasing tomorrow we right out to the first of the curative, which is minus X minus one x minus two, divided by three x. So if prying, because a zero gives us xy costa one or etc cause or two. So we have the reserve intervals from 0 to 1 no 1 to 2 in the front or to infinity on the first interval. If prime is negative, so the function is decreased on the second place. If prime is positive, the functions increasing on the last interval if primaries negative again. The functions de quizzing. So that means Naser there is a local I mean het xy close the ones. So if one, um, you cost over 5/6 in the less a local minimum on local next month at X equals 22 the value will be left to close to four with reminders to third normal too. For puffy right out The second of narrative which is two minus X squared, divided by three x square. So if the prime because of zero we have X equals two class minus two. But yes, minus put off to is outside of the men so we don't need to consider this point. So we have to stop intervals from the old to write off two in the front route off to infinity on the first interval Double primaries, Politics of the fortunes country of our who was thinking about the experience Negative. So the fund your contract down And then there's the inflection points. That exit was the route off to Now we are really the graph. So the function pony Define bones zero to infinity. Um so firstly come is decreasing can keep up. They hit the, um, reflection point at X equals the rooftop to So you hit the, um X equals toe one, which is a local Milliman noted inflection points. So then the cup that no function is increasing, But when you increase, you change it. Contiguity is here, so there is a reflection point and, uh, he hit another local extremely, which is a welcome 80 minute X equals two the staff people issue. So this is a sketch of the graph.

For this question with in party we look for where? Everything for the most sympathetic behavior. So full of verticals and Celtic ways. See that the domain off if is from zero to infinity. So we look for this limit. X coast is here from the right and we find out this actually caused to infinity. So there is a There's a vertical sympathetic at, uh a little goes a Metallica XY close to zero in the for the horizontal sympathetic revalue, its limit exposed to infinity and the extra cost of minus infinity. So there's no no heretic, no horizontal sympathetic in this case for part e for the increasing and decreasing tomorrow we right out to the first of the curative, which is minus X minus one x minus two, divided by three x. So if prying, because a zero gives us xy costa one or etc cause or two. So we have the reserve intervals from 0 to 1 no 1 to 2 in the front or to infinity on the first interval. If prime is negative, so the function is decreased on the second place. If prime is positive, the functions increasing on the last interval if primaries negative again. The functions de quizzing. So that means Naser there is a local I mean het xy close the ones. So if one, um, you cost over 5/6 in the less a local minimum on local next month at X equals 22 the value will be left to close to four with reminders to third normal too. For puffy right out The second of narrative which is two minus X squared, divided by three x square. So if the prime because of zero we have X equals two class minus two. But yes, minus put off to is outside of the men so we don't need to consider this point. So we have to stop intervals from the old to write off two in the front route off to infinity on the first interval Double primaries, Politics of the fortunes country of our who was thinking about the experience Negative. So the fund your contract down And then there's the inflection points. That exit was the route off to Now we are really the graph. So the function pony Define bones zero to infinity. Um so firstly come is decreasing can keep up. They hit the, um, reflection point at X equals the rooftop to So you hit the, um X equals toe one, which is a local Milliman noted inflection points. So then the cup that no function is increasing, But when you increase, you change it. Contiguity is here, so there is a reflection point and, uh, he hit another local extremely, which is a welcome 80 minute X equals two the staff people issue. So this is a sketch of the graph.

All right. So we are first being asked to find the vertical and horizontal ask until so, um, dysfunction can be rewritten as one over e to the X Square. And now, if you haven't already noticed, um, this bottom function, I'm dysfunction denominator cannot equal zero because that's what defines vertical. I think that's one of the ways. And since exponential function never equal zero, there is no vertical Jacinto. So no vertical ascent toe. And so now we gotta find the horizontal axis until we find that I find the limit. So the limit is X goes to infinity, and this will be one over e to the X squared. And as you can see as, uh, this number guests, uh, the denominator gets exponentially higher. You get one of the infinity, which is a one of a really big numbers, just zero. And this is a case also for negative numbers because negative numbers squared. It's the same thing as positive number squared. So this was also a plus or minus infinity. So we have a horizontal attitude at y equals zero. Yeah, and then now, to find the intervals in which function increases or decreases, we apply the first derivative cath that I've been taking the first derivative. In this case, you will have to apply the chain rule, so I'll come out to be negative. Two X equals negative. Two x 10 e to the negative x squared. Uh, sorry about that. Just bring this down. I was just negative two x or eat the minus X squared and this can be rewritten at negative two X over E to the X squared. Then you set this equal to zero. The denominator cannot equal zero, so that's not one of the critical number. Negative two X equals zero when X is equal to zero. So we have a signed chart evaluation here, so it's going to be X, and you put it at zero, and then you bring it down and they were looking at the sign of a crime. But people are numbers less than zero. You get positive numbers and you fucking greater than zero. You get negative numbers, so we noticed increasing and decreasing. So we have a local max occurring at X equals zero. Uh, and we know that it is increasing from negative infinity to zero, and it is decreasing from zero to infinity. Now, to find where the funk says, uh, find the functions can cavity would take the second derivative tests that they have been trying in the second derivative. So again, this is a little bit more chain rule and, uh, combination of some protocol, and this will come out to be negative one plus two x squared statistical zero. Um, this part of the function cannot obviously equals zero. So we have to set this part of the pumpernickel zero. So this will be negative one plus two x squared because you have to find a critical number. You add one divide by Tuesday. You got X square called 1/2. This will be X equals. The square root of one is one. So we one over root two plus or minus because we took a square root. Now we can do a sign chart evaluation. This will be negative. 1/2. It would be positive one over. Richer, not a line, but not a line. Looking at the sign of a double prime your fucking values less than negative one or two. You get positive numbers between these two negative and positive. Still can't give up. Down, up. So we have a con cave up interval occurring from negative infinity. Two negative 1/2. And from one over to to infinity. We have concrete down occurring between negative one over to positive 1/2. And we have inflection points occurring when sign change occurs. So that's occurred that both plus and minus route to I mean plus or minus, uh, 1/2. So inflection point point occurred that plus or minus 1/2. Now we have enough information to draw a graph. So it looks something like this. Uh, you know that there is a local max at zero, and we have a concave up shape from here from the negative. So we're kind of coming up like I use so this is like, like up you, and then I'll turn into a concave down shape. So now we start decreasing after zero, you come down and and then I'll go off to zero. And this this max occurring a gray here. Um, the peak is supposed we had zero Mhm. Sorry, if that wasn't clear enough, but this is the graph of F WebEx and it's symmetrical. That's it.


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