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Problem 56 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) = e^{\arctan x} $

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Video Transcript

Okay, so we're being has to find a vertical and horizontal passenger effort back. And so for vertical hasn't. So we're looking at a fucking that's e raised to the power of the arc can of axe and is an exponential function. And it is continues everywhere and and have nose, um, sort of vertical. Ask himto anywhere. Is that possible? I mean, it's not in this case, so we know that there's no vertical acidosis, so no vertical ascent because it's an exponential function. However, for the the Ark can function. We have. Ah, we do have a scent of it on the Ark Chan funk and show as a reminder. Um, the graph, actually. Ah, looks like this has, like a pie, and it kind of goes like this. This goes off to pie over to andnegative pie over, too. So this is pi over to. This is not very drawn very well, but this goes off to negative pie. So there's a There's an acid, there's a horizontal assets, a recurring and art panics. And there's a very good chance that it's also occurring in F of X. So we're going to look at how the function behave as they go off to infinity. So the limit as X goes to infinity. So what we're going to do is first going evaluated for our ten. So our can X and you'LL see why in a second and to the limited excuse Intramural Panamax Ahs! You've seen this picture is pi over too? And so that means that the limit as X goes to infinity of ethics, it's going to be e to the pie over too. So we do have courses on things and for the limit as explosive. Negative infinity for our cannon back is going to be negative. Pile over too. So that means that the limit as X goes to negative infinity of FX he's going to give us Yeah, today I get to fly over to so we have two horns on the after we have won it. Why the e um anything that down here? Why t e ty over too and e to the minus pirate too. To find where the function is increasing and decreasing, we take the first planet First derivative testament. You find sir a prime. And since this is an e racy or can we're going to have to do, general. And so it comes out to be our can of X all over one plus x word. And now, if you look at this function, we know that the exponent the function is always positive, and we know that the denominator cannot be negative so that dysfunction is always zero is always greater than you know. So it is always increasing, always increasing, I said. I mean, it is increasing from negative infinity to infinity, and that is so That means they were so senses, always increasing. We know that there's no local minimum max either, because there's no point in which there is the function of dipping either. So there's no local Miramax. No, now for controversy. We look at the second derivative and see how f double prime changing that's double prime of X is equal Tio e. R. Ten of times negative two x plus one all over one plus X squared. Where'd you think? Easy. Jochen can never equal zero. So we have to look a negative Force one because, you know, and to cross one zero and you get X equal one half. Now we do a sign chart and we're going to evaluate it around one half. So if we look at the sign of af double prime when it is less than one half of this positive number is greater than one half its negative numbers that it's conking out, I can keep down. So it's Kong cave of from negative infinity to on half. And then Khan gave down from one half one have to infinity So and also we have an inflection point. You have a deflection point occurring at one half because that's where the sign change for our con cavity is a crime going from positive to negative. Now we have enough information to draw our graph us. So this is going to be all this is going to be all above the X axis because the lower bound for our, uh where is aunt asking those eating negative private too, which is still a positive number. So this could represent a two five two. And this could be you're a pilot too, So the function will actually look something like this that will be coming up. And then one half it was switch and have a new con company. So it goes from conclave up, two down and that's basically the graph of F Back like that. This is Justin more straight and no local maximum and as it