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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)=1+\frac{1}{x}-\frac{1}{x^{2}}$

A. vertical asymptote: $x=0 \quad$ horizontal asymptote: $y=1$B. Increasing on interval $0< x <2,$ decreasing on intervals $x<0$ and $x>2$C. Local maximum $=\frac{5}{4},$ local minimum does not exist.D. Intervals of concavity: $x<3,$ concave down, $x>3,$ concave up. Inflectionpoint at $\mathrm{x}=3 .$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

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

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in this question. We have five different parts, so first is the sympathetic. So for horizontal sympathetic, we need to evaluate the foreign to limits. X goes to infinity off a fix. Um, which is one and the X goes to miners in divinity off eight weeks. It's also what? So there is a horizontal sympathetic why it goes to one and for the vertical Sympathetic Because we can see that if X equals zero FX, Uh, it's not defined it. So we have one vertical symptomatic X equals zero because we we have something divide by zero. In that case, okay, s So this is a synthetic in the for a second part for the increasing and decreasing turbos, we need to find his first of the curative, which is two minus x over X cubed. And they're just let, um this first of the director because they're all so we have xy constitute. That means we have on the following in her rose. Um, actually, we have the right because X equals zero. It's not in the domain, so we need to consider there's, like, minus infinity to zero in the 0 to 2 in the two to infinity. So for the first in general, if prime less than zero say it's decreasing the second turbo if prime is greater than zero. Safety increasing last interval. If prime miss less than zero again decreasing. We don't have the There's no no co Maxime Miss Case, because X equals zero is not in a domain, but there is a local maximum. It takes at X equals 22 with Valley Way for two. Um, half of two equals 25 for four. Okay, so this is posse in the folk Hadi. We need to find second a Do it first, which is two times X minus three, divided by extra before in the legs. The second a charity because is here we have X equals three again. We have three intervals minus infinity to zero 0 to 3 in the three to infinity on the first interval. F top of prime less than zero. So it's down work concave downward and the front 0 to 3 after about clients also zero. So it's again down worked and the last one if the other privatise positive. So it's upward. So we have one inflection points at X equals three now we are ready to Ah, get off this information. The graph. I mean schedule graph off the So we said have according a system. So this is why pick sixes, you know, and we labeled horizontal sympathetic. This is why it cost to one. And there is ah, vertical sympathetic, which is just what exists. So look. So the graph looks like this Not here is conclave down and the decreasing and from 0 to 2 is increasing up to five for four here, up to this point. And these also come caved. Um, it's become kept on you and your ex are equals 23 something magnets. So there's an inflection point here, So the can cavity is that you can cavity is changed. It's changed at at this point and then we have a local Uncle Max here, and this is the inflection and that we have horizontal symmetry are critical. Sympathetic with Richards waxes

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