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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 ( 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 asymptoteB. Decreasing:$0< x <1$$2< x <\infty$Increasing:$1< x <2$C. if $\mathrm{x}=1$ then $\mathrm{f}(\mathrm{x})=1-\frac{1}{6} \times 1^{2}-\frac{2}{3} \ln 1=\frac{5}{6} \approx 0.833$if $\mathrm{x}=2$ then $\mathrm{f}(\mathrm{x})=2-\frac{1}{6} \times 2^{2}-\frac{2}{3} \ln 2 \approx 0.871$D. $(0, \sqrt{2})$ concave up$(\sqrt{2}, \infty)$ concave downInflection point $=\sqrt{2}$E.SEE GRAPH

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) Find the vertical and …

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For this question in part a we look, we are looking for the synthetic behavior, so full of vertical symthe see that the domain of f is from 0 to infinity on. So we look for this. Limit x goes to 0 from the right and we find that this actually goes to infinity. So there's there's a vertical ambato at vertical symbo x, equals to 0 point and for the horizontal sympathetic we evaluate limit x goes to infinity and it actually goes to minus infinity. So there's no vertical, no horizontal symptotic in this case for part b for the increasing and decreasing interval. We write at the first of the derivative, which is minus x, minus 1 x, minus 2, divided by 3 x. So if prime equals to 0 gives us x, equals to 1 or x equals to 2 point, so we have 3 sub intervals from 0 to 1, from 1 to 2 and from 2 to infinity in on the first interval. If prime is negative, so the function is decreasing on, the second interval is f. Prime is positive. The function is increasing on the last interval. If prime is negative again, the function is decreasing, so that means there's. There'S a local minimum at x equals to 1. So f 1 equals to 5 over 6 and theres. A local minimum and local maximum at x equals to 2 on the value. F. 2 equals to 4 over 3 minus 2 thirds normal 2. For part b. We write of the secondrate, which is 2 minus x. Squared divided by 3 x squared so if double prime equals to 0, we have x equals to plus minus 2 point, but since minus root of 2 is outside the domain, so we don't need to consider this point. So we only have 2 sub intervals from the ear to root of 2 and from root of 2 to infinity on the first interli double prime is positive, so the function is concave on the second, the pens negative, so the functions, concave dome and then there's the Inflection point at x equals to root of 2 point now. We are ready the graph, so the function only defined on 0 to infinity. So first it is decreasing. Concave up, then you hit the inflection point at x equals to root of 2 point. Then you hit x equals to 1, which is a local minimum, not an inflection point. So then the the function is increasing, but when the increasing changes concavities here, so there is a infraction point and the heat another local extreme, which is the local maximum at x, equals 2. Then it starts decreasing its. This is a sketch of the graph of it.

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