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University of North Texas

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Problem 22 Medium Difficulty

Use a computer algebra system to graph $ f $ and to find $ f' $ and $ f" $. Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of $ f $.

$ f(x) = \dfrac{3}{3 + 2\sin x} $

Answer

1. $-2.124+\pi k$
2. $-1.017+\pi k$
$k \in \mathbb{Z}$

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Video Transcript

we want to use some kind of cast software to graft the function. F of X is equal to 30 over three plus to sign of vets and use this software to find the 1st 2nd derivatives, wrapped those as well and then estimate the intervals with punctures increasing, decreasing interval con cavity and find any extreme values and points of an election. So I would have had a graft f of X here. And if you were to really zoom out further along the X axis, you would see that this is really periodic and it has the same shape that repeats over and over again. So which makes sense because we have side of exile. We know side of exes, periodic. So for what we have here, it's going to repeat like this every pie. So if we go ahead and just described this over the interval negative three pie half step I have. We know we can go ahead and just shift our interval by pi each time. And it will give us our new intervals where the functions increasing and decreasing in any con cavity. So let's just go ahead appear and right period of this is going to be pie, so we know we need to add plus Pike A at a time. Right now. Um, I went ahead and already graft f private, back soon and again since I know it's gonna be periodic and repeat between negative three, perhaps the pie half. So I just want to head and graft only that part there. So let's figure out where the smoke is increasing on. Dishonorable. So remember, we know that prime of X greater than zero is when the function will increase. So that looks like it's going to be from negative bleep. I house two negative pie. How And then we could go ahead and transformed this that we add plus pi k on each end to give us negative three pie, huh? Plus pie. Okay. Too negative. Hi. How plus pi k And remember? Okay. Should just be some intern. Sure about the remember editors are 01234 and negative. 1234 So now is go ahead and find where fbi lamex. This is strictly less than zero or where it's going to be decreasing on this interval here. So we're going to end up with So it looks like it becomes negative. Negative pie, huh? And I'll stay negative until pi up. And on that interval, that's the only place where, after prime of excess sugar, less than zero. So we're gonna do the same thing that we did before Ad pi k to each of our in pieces. So plus pie. Okay. And that will give us the new interval. Oh, negative pie, huh? Plus pie cake and pie, huh? Plus pie. Okay. And again, que is just some element of the intruders. All right, so we know where the puncture is. Always increasing and decreasing. So now let's figure out if he's going to be local backs is our local men's. So over here. Negative three, Perhaps to the left of it. The function is decreasing because a crime of exes lessons and to the right of it is going to be increasing because Alexis positive. So that means at this point here, what kind of a local men at negative pie, huh? Well, it is increasing and to the point and then decreasing after. So this is going to be a local Max and her pie half over here. Well, it should match up with what? We have a negative 35 house because it is going to be periodic. So little left, it is decreasing to the right in his increasing. So here we have a local and we can write this little bit more generally if we want that. So I'm gonna call this here the local backs one. So we know we have a local max. Uh, X is equal to negative pie half plus pie, okay. And our local men's local men, Pat X, is equal to, and I'm just going to use the right in point here. Negative pie, huh? Or just negative pie. Have a not negative and plus pie get. And if I were to do K zero negative one, I would get our other point here. All right, so we know where the function is increasing and decreasing for all this intervals, and we know where all the local taxes and local men's will occur. Now, let's go ahead to our second derivatives. So we confined points of inflection as well as any point where the function is, uh, increasing, or are any points of the election and where the focus is concave up concrete down. So I wanna have a graft that interval once again. And so let's first figure out where the interval for our period that we're working with will be positive and negative, and then we can see if we can generalize it from there. So remember, Yeah, Double Private X, strictly larger than zero. Tells us our bunker is going to be calm came up and at least not negative three pie halfs to pie hat that's going to occur at. So we're going to include our point this time. So negative three pie. How too Well, our first X intercept, which is negative. Two point on to four. And this we're going to exclude because prime of X, it's not shoe collection, that zero there. And then we're going to union this with, um from our next X intercept negative 1.17 until the end of our period, which is pie and we want to include are all right. So if we were to go ahead and do the same thing that we did before, we can go ahead and right this as so we still want to shifted by pi. So let's add, plus pi k so plus pie. Okay. And that's going to give us negative three pie half, including the end point plus pie cake to negative 2.1 to 4 plus pie. Okay. And then we exclude that in point, and then we union it with negative 1.17 Add up. I ke that. Now let me go a little bit and then pi half plus pie, okay. And then remember this too then que is just some arbitrary integer. So any of these intervals here will be where the function is. Com Cave up. If you wanted, you could try to find some nice way to combined these in points. And this should be a bracket on. And here, since we are including pie half since at that point it's still, you could try to find some nice way to make this just one in trouble if you really wanted to. But I would say, Let's just go ahead and leave it like this. All right? Now we can go ahead and find where private Becks is going to be stricken. Lessons are on this interval here. There's going to be con cave down. And for us that only Walker on negative. 2.1 to 4 to negative 1.17 And just like before, we're going to add the plus pi k so we can find all of our intervals. And that's 20 of us. Negative. 2.1 to 4 negative. 1.17 Almost rejected. Put the plus. Hi, Kate. We're here plus Hi, Kay. And negative. 1.17 plus high. Okay. And Kay is just some element of the intruders, as though this tells us all of our intervals where the function is going to be con cave down. And now let's find our points of inflection. So on this interval, here we have our Exeter subset Negative 2.124 So, to the left of this, the puncture is going to be calm. Keep up to the right is going to be calm. Keep down. So we have this change of con cavity. So this will be a point of inflection and our other X intercept here. Well, it's conclave down toe left conclave up to the right. So this will also be a point of fracture so we could go ahead and write this more generally as so this first point of inflection. Here it's going to be negative. 2.1 to 4 plus pie cake and our other put inflection negative 1.17 plus pi k and again, que is just some element of the intruders. So this describes all of our points of inflection, and we have our two intervals here for where it's gonna be called Cave up in calm paid down. So we found all points and reflection where the functions always conquered, conquered down all extreme values and all intervals with the focus is increasing and decreasing, So we've done everything we need to do.