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Problem 55 Hard Difficulty

(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) = \ln (1 - \ln x) $

Answer

$f(x)=\ln (1-\ln x) \text { is defined when } x>0 \text { (so that } \ln x \text { is defined) and } 1-\ln x>0 \text { [so that } \ln (1-\ln x) \text { is defined }]$
The second condition is equivalent to $1>\ln x \Leftrightarrow x<e,$ so $f$ has domain $(0, e)$
(a) $A s x \rightarrow 0^{+}, \ln x \rightarrow-\infty,$ so $1-\ln x \rightarrow \infty$ and $f(x) \rightarrow \infty \cdot$ As $x \rightarrow e^{-}, \ln x \rightarrow 1^{-},$ so $1-\ln x \rightarrow 0^{+}$ and
$f(x) \rightarrow-\infty .$ Thus, $x=0$ and $x=e$ are vertical asymptotes. There is no horizontal asymptote. (b) $f^{\prime}(x)=\frac{1}{1-\ln x}\left(-\frac{1}{x}\right)=-\frac{1}{x(1-\ln x)}<0$ on $(0, e) .$ Thus, $f$ is decreasing on its domain, $(0, e)$
(d) $f^{\prime \prime}(x)=-\frac{-[x(1-\ln x)]^{\prime}}{[x(1-\ln x)]^{2}}=\frac{x(-1 / x)+(1-\ln x)}{x^{2}(1-\ln x)^{2}}$
\[
=-\frac{\ln x}{x^{2}(1-\ln x)^{2}}
\]
so $f^{\prime \prime}(x)>0 \Leftrightarrow \ln x<0 \Leftrightarrow 0<x<1 .$ Thus, $f$ is $\mathrm{CU}$ on (0,1)
and $\mathrm{CD}$ on $(1, e) .$ There is an inflection point at (1,0)

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Video Transcript

okay. To find the vertical and horizontal ascent of effort back, we have to personal. We have to examine the domain of the function. So the national companies are all X rated Indio. So the inside on minus Alan X's graves and Jiro you get one is greater than Eleanor Vex, and you take E on both sides and that is greater than Excell Access Western E sort of domain is essentially from zero tio. That is the domain of our function. So what does that tell us? Well, we know that this is a natural like furniture we notice continuous between zero you. But it doesn't mean that there could be a vertical acids are occurring at the imports. So we can evaluate that by plugging in that limit the limit of exit go studio FX. So if you plug in zero, you get national level one and international art of zero is negative Infinity. So you get negative to the positive. So naturally, one thousand phonies Incredibly, Which is it going to Trinity? So there is a ah vertical hospital recurring and X equals zero, and it goes off the positive meaning if you got a e from the left of effort. Beck, you get natural log one minus. And at National Aga is just for one. So one minus one is zero national log of zero gives us negative infinity. So that's also going off to if every so we have a X equals zero and e for a horizontal. Lassiter, Um, there is done because we have such a strict or maine. So she take eliminate excess in Trinity, who we've technically already evaluated tea and point. So no words are to find where the functions increasing, decreasing. We have to find first derivative test. No problem. This will come out to negative one over X times one minus our Next we looked for the critical numbers of dysfunction. So that's where the bottom could possibly equals zero and so e we know that it could happen at zero, which is already part of our domain. Still, we'LL evaluate it, Teo. And in one minus l. A durex that occurs when Deccan equals zero when it is e so is it was the act which again is the end of our domain. So we're going to die away from zero to a and we don't care about anything outside of that. Ah, you fucking value between Juni get negative numbers. So we know it is decreasing the entire time on its interval, decreasing for all the t e. There is no local maxim and because there's no change and the direction of the function, so to find the con cavity of the function would take the second derivative and that comes out to be negative. Allan of X all over X times one minus Alan Rex, this equals zero and exes Ical too one. And we again on critical number. That kind of your own e. So we're going to evaluate it at tio ese are signed chart. It's going to happen around zero one and e and we ignore everything outside of it. So between your number zero on one, you get positive numbers and repugnant Other shouldn't want need and negative numbers. It is currently rub down. You can take off from zero to one, and Kong came down from Plan Tio. According to D, Now we have we have to have an inflection point occurring at X equals one is a tactical one. In fact point Ah, now we have enough information to draw a graph of the function, and we know that it is the limited between June and Jesus will be like somewhere over here wherever I set that to be and it is decreasing the entire time. So it's coming. Remember when As limited excess dealer who's going to positive for? So we know it starts up here. It was going to come down, come down, come down, have a conclave up ship until one. So it's going to kind of have a U shape, but at one and switches and it actually goes to a con cave down shape and it goes off to negative infinity. It goes like that. This is this is e here and just basically the crab. And this is, uh, zero. This is that one, and this is a graph of Catholics.