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(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 $

a) Vertical asymptote is $x=0$

There is no horizontal asymptote

b) Decreasing: $0 < x < 1$ $2 < x < \infty$ Increasing: $1< x < 2$

c) if $x=1$ then $f(x)=1-\frac{1}{6} \times 1^{2}-\frac{2}{3} \ln 1=\frac{5}{6} \approx 0.833$

if $x=2$ then $f(x)=2-\frac{1}{6} \times 2^{2}-\frac{2}{3} \ln 2 \approx 0.871$

d) $(0, \sqrt{2})$ concave up

$(\sqrt{2}, \infty)$ concave down

Inflection point $=\sqrt{2}$

e) SEE GRAPH

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Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

but to find the vertical and horizontal atom tones of Africa. Ah, oui. There's a couple of ways we can do it. So previously we've been doing it by looking at the denominator and where that equals zero to give us thie. Very classical. But another way to do it is to take the limit off some number A, and it should give plus or minus infinity. Then we have a very flat. So we know that first of all, that there's a national documents Foxes. So the domain of dysfunction is all excreted D'Oh So it would be interesting to evaluate what's going on at as limit as X goes to zero from the right and you can do to accept a Titian. So you get your own minus zero minus. And at national art goes to infinity. It goes to negative infinity because reminder that the graf the national art function looks like this like this and it goes off to negative finished as it gets to zero. So nave negative makes in positive infinity, which is still a very classy So So we have you really blasted out a tactical zero, and if you know Paul in over farm for international continuous and everywhere else honest domain. So we don't have to worry about anything else. This is Theo, only radical ass and talk to her now. Ah, to find horse out. Asked that we take a limit at that grows to infinity. We want the word about negativity because remember coming greatest zero eso this one. You also have to do a kind of an intuitive approach. It is Otherwise we stuck doing some sort of horrible local tall problem or a very complex, very complex limit so you can think about is and finish minus infinity minus some number because national all kind of dies off. So infinite, even infinity, That's what What that really gives us still some some form of infinity. So we don't really We have no horizontal ask himto so no orders our class until so now what we do is we're going to find the interval than most folks and increases and decreases. So we take the first derivative and apply. And first, first it just comes out as one minus X over three minder to three x. We start the sequel to zero. So what I'm going to do is going to bring two of the three acts over, and then I'm going to simplify attraction. You ever get out? I'm gonna be right there as three minus X over three, and then I'm bringing to over three x over. And then I'm gonna multiply both sides by three x and I should give me nine X minus three X squared and it canceled out with three B And then when I got about three on the outside, I get six here. I mean, I multiply by three combo size. I get six here, and then I bring six over our nine X minus three X cleared minus six equals you. Hey, I pull out a factor of three, and that's what you mean. Three, um X minus X. Where'd my net two and there's equals zero. And this could be him. But this commune factor at explain it to times one minus X giving us the solutions. Actually, ical two and one. Ah, now we're going to do our sign tried evaluation that will have what's evaluated zero because our domain is greater than zero. So we want to see what's going around there. And one and two bring a line down. Everyone. I ignore all that lesson. Joker, That's not not made. And now we find the sign of the primes are between June one. You get negative numbers. Ah, between one and two. Get positive numbers and all greater than to get negative number. So we have a decreasing increasing, decreasing. So we have a local men at tactical is one and a local Mac at X equals two. To find the con cavity of our function, you have to get thie second derivative and applied a second over. Ted and this second David come down three negative three squared six over nine x Clint and he said, the topic or zero. So you say calls. You know, So you have negative three X squared plus six equals E o. You can get rid of the negative of the plus minus six will be three x squared equal six divided by three. Got actually cool, too technically plus or minus. But we ignored the negative because all our numbers positive remain. That's credit and zero. So I would do another side chart evaluations. So we have zero and two. I like a double prime. We ignore last zero. So you plug a number of Liston in between two and we're two positive number. You plug in a power greater than retreat and negative numbers. So you have conquered up down the conclave of between two hundred two hand Kong came down from negative infinity to Oh, no, not not Trinity. Sorry about that. We're two to infinity now. We have enough information to draw hope. I also forgot about the inflection point. So we have an inflection point. It occurs when a sign changes and con cavity occurs. So that's between it goes on record that we're too that ex Eagle who to now we have enough information to our craft of affect So we know that it is restricted by this domain. So it has to be X rated video. We have bor. We have no horizontal jacinto and we notice con cave up So we know that there's a U shape occurring So it's going to come down to the glass enter as Jiro So it's going to come down like this. You have a u shape. We're going to create a little cold men at one when they keep going up and then we're going to Sweet con camp. There were two which is approximately right here. There's gonna increase between one and two. So we're going to get a local max at two. And then I still can't give down now and we know has been a decrease after, too. So it's just going to go down. So this is two. And this is what? And that is it. That's the graph of athletic