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(a) Graph the function.

(b) Explain the shape of the graph by computing the limit as $ x \to 0^+ $ or $ x \to \infty $.

(c) Estimate the maximum and minimum values and then use calculus to find the exact values.

(d) Use a graph of $ f" $ to estimate the $ x $-coordinates of the inflection points.

$ f(x) = (\sin x)^{\sin x} $

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Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

So first we want to graphic the function of X is equal to side of X rays to the side, Lex. Then we want to explain the shape of the ground by computing the limit as X approaches zero from the right and as X goes to infinity than in part seeing, we're going to estimate the max and men's from the graft Lee made in part eight. And then we want to use calculus to find those. Exactly. And then lastly, we want to graph second derivative to estimate the X coordinates of inflection points. So I went ahead and already graft, uh, f of X and you might notice that I only really labeled this first till day kind of looking shape. And if you were to graph go in out on either side, it just repeats this till the shape over and over in forever. And you might notice that the next Tilly starts to pie and that is going to end at I didn't write this on here. Political right Ri pi would be this under endpoint, so it kind of looks like these till days are periodic with two pie. So if we just figure out everything for this first till day looking around, then we will know all of the values for been on either side. So let's just go ahead and write does not really quickly. So our max is are Actually we were just also find what the maximum that's art was already did. Uh but we will need to know that this is periodic to pie for when we're finding the bodies exactly apart see as well as estimated our inflection points. So once we find those just on this first period here, we know we could just add two pi k were case of integer to get the rest of our solutions. So now, for Part B, uh, let's talk about the behavior as X goes to infinity. So just by looking at the graph, you might notice that it since it is periodic, it never really gets closer to any value since the values just keep repeating. So from that we can conclude that side of X to the X. Is this going to diverge? So socko idiot really close to anything. And then if we look at the graph here, it looks like that released by the graphing calculator that I used says that the value 40 should be one. So let's see if we can actually show that the limit as X approaches Zero for the right is going to be so and something I'm just gonna do is move this up into this corner right here. So I could just work down as opposed to having to go sideways. So let's just go ahead and implied this director first. So if we were to just apply it directly, we'll sign him. Zero is defined, which is just going to be zero. So it's gonna be 0 to 0, and I have no idea what that means. But any time we have something two 0 to 0, we can use E and natural log to rewrite it. So let's go ahead and do that. So limit as XO perches on the right. So I have based E now and then I'm going to take the natural log of sign of X to the sign of X, which I can rewrite this sign of ex natural log of sign of X and again Mrs got him applying the limit because maybe we get something that works. A little bit better. So this is going to be e to be. We'll sign of Mexican, goes to zero natural log of something Going to zero is going to be negative Infinity and again, I don't know what zoom type of psychic ability is. So let's go ahead and rewrite this because any time we have zero times infinity, that means we can rewrite to this to use local cultural. So the limit, as expected from the right of e. So I'm gonna rewrite this as one over sign of X, and then that would be like I'm dividing it. But that's the same thing, is just dividing by Cho. Seek it. So I'm gonna rewrite this expression now as the natural log of sine X over. Go seek it. Thanks. And then just one more time. Let's go ahead and apply this So again, our new mayor is gonna be negative. Infinity, And then our denominator is going to go to end Remember Coast to get just one sign And since we're approaching from the right, all the values for a sign should be positive. So we have made a penny over infinity. So that means let's supply local cultural. So this year is going to be equal by local calls to the limit as X approaches zero from the right of E So the derivative of natural lock side of X, it was gonna be won over sign of x times the derivative of our inside, which is a sign of X So co signed X and then this is gonna be all over. You know, I skipped my feet a little bit and the derivative of co secret is gonna be negative. CO c can't cope tangent. Now notice that coast side of ex oversight of ex Well, that's co tablet. So in our numerator, we already have co tension. So the co tangents well counts out and then we could rewrite coast. He can't as or one overcome seeking as side. So you know how to really be sent to the limit as experts zero from the right of the to the negative. Go seek it x for negative sign X since we went ahead and reciprocated it again. Now we've got unemployment again because we know side of excuses. Zero I was going to the limit as Ex purchase. Or so since we apply it But then it goes away and it's gonna be raised to the zero, which is just. And then that also explains the behavior as, um sign goes the pie or any multiple of pie. All right, so we went ahead and show that So let's go ahead and find our maximum and values. Exactly. So to do that, remember, we're going to need to find the derivative of this craft, though, Um, any time we have something like this, we can still use that exponentially ation trick every bite it into something that's a bit easier to work with to take the derivative. So let's do that. So I'm going to rewrite this again as the sign of ex Natural log of signing checks. Now let's go and take the derivative of this so f prime of X. So we're going to have to take the derivative of There's using changeable. Remember change Will says we take the derivative of our outside function, which is just going to be needs of X, and the driver will be intellect city to next, so that doesn't change it all. So sign of X Times Natural Law of side of X. But then we need to take the driver inside function, which is a sign of ex natural log of Sign of its. And then I'm just gonna rewrite e to the sign of exceptional Santa picks as sign of X through the sign of X. Just save a little bit of space. And now to take the derivative of signing next time's natural outside of extra going to need to use product. Cool. So remember, product Will says we need the 1st 1 So it's gonna be side of X, then times the derivative of natural log side effects, which we already found over here. Toby. Just co tangent of X. So let's go ahead and write that down or very obliterated as co sign of X over signed checks. And then we're going to add this too. When we switched the order of these, that was gonna be natural. Log of sign of x times, the derivative of Sina Becks, which is co sign elects. So now notice that here the sign of X's cancel out and we can factor on a cool side of X. So let's go ahead and do that. So sign of X raised to decide of bucks times. So we factored out that co sign of X on the inside. So we have co sign of X here and then a factory that out, We'll have one plus natural log of sign of X party. Now, to fight our critical values, you're gonna want to separate C zero. And if we go and look at our original graph again, we could see that sign of X to the sign of Ex Never become zero at any point, so that will never be zero. So when we apply the zero product property, the only numbers are the only things direction you say zero co sign of X, add one plus natural log Sativex. So let's go ahead and do that off on the side. Here. So we have co sign of X is equal to zero as well as one plus natural log of side effects is equal to zero. So co sign of X is he was a row that pie half three by halves and really anything pie half plus pi k. So it's got it right that so we know ex physical t pie, how plus pie. Okay, and everyone should just go ahead and plug in pie have and to hear doing that will give so of pie. How? Because Remember, if I add pi k to this, they really should just give me the sane, uh, answers and doing that it will be, well, Zain a pie. How it is one and sign again. It's gonna be one. So it's 12 The one or what? And if we were to go and look, we have pie half the first power. Hi. How and we get our output of one. So that matches up with what we would expect. And now we can go ahead on and solved over here for our other values. And doing that will give us So a person truck the one over get natural log of design a box. Is he getting negative? One exponentially ate each side, giving us a sign of ex busy t e to the negative one. And then we would take our HQ side off This So we get one solution that's going to be X is equal to Arc side show. Just like this. A sign papers sign in burst of e to the negative one and we get a second solution being exes it into pi minus sign and burst of me to the Remember we get that other pie answer from if we were to just draw your circle right here. So if this is where you need to, the negative one is over here would be another solution for so they end up with those two values and they should give the same I'll put for us. And now we could go ahead and figure out what this value should be exactly, since they should have the same one that we could just go ahead and plug it one of them. So, uh oh. Side and bursts of e tonight, the first power. Well, this is going to be will sign of sine inverse will those just counts out when we have you to the night of the first power. And then same thing. Our power here. So you have each of the first power that raised you to the negative first power. So that would give us our actual minimums. And then again, if we wanted, we could just go ahead. Like I was saying at the start of the video ad plus two pi k do each of these values here, and it would give us all of our answers for where all of our minimums are going to occur. All right, Now, for the last part, Um, so we're supposed to graph the second derivative, and as you can see here, that's pretty long. And it would make the video, uh, much too long. So I just went ahead and found the drip. No, already using a derivative calculator. And since we've already used the derivative calculator for some of the earlier problems, I really did it have much problems with my conscious of just plug in this into a derivative calculator to find that. And after I did that, I would have been grafted. And I found just the ex intercepts odd the first period. So from 0 to 2, pi earned from zero to a pie, which occurred at about 0.95 and 2.2. And so we can see that on either side of these. So to the left of 0.95 the function is going to be conquer off because after bill promise trigger larger than zero and to the right of this is going to be con cave down since half double private, strictly less than zero and then to the left of 2.2. It's complicated, so each of those do end up being points of inflection. So we know that at 0.95 well, Avlynn and about 2.2 well have points of inflection. But then again, we're going to add plus two pi K to all of our answers here because of the period ISS ity of this. And so just remember again, this K is just a element of the intruders.