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University of North Texas

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Problem 2 Medium Difficulty

Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = -2x^6 + 5x^5 + 140x^3 - 110x^2 $

Answer

$x=1,3$

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Video Transcript

you want to produce graph of F that rebuilds all of the important aspects of the curve where F is equal to negative, too. Times X to the six plus five times X to the fifth plus 140 excuse minus 110 X squirt. And what we want to do is to use the graphs of F prime adaptable prime. Do you estimate intervals over the functions increasing, decreasing, binding the extreme values, intervals of con cavity and inflection points. So just for the sake of brevity, I went ahead and already wrath all the functions we're gonna be using now or on the left. This is just the graph of FX. When I just plugged into my calculator without messing with the window, it'll and you can see all the information between 12 about five seems to be lost. We really don't know what's going on up there. And also between zero and one, everything seems to be really squished together. So maybe we can find some nice interval to blow this up so that we get all the information. If not, maybe we can get one that has everything in it. And then one smaller viewing rectangle as well. So I went ahead and found the first and second derivatives. And remember to find these, we would use the Powerball and I went ahead and plotted these. Now it may not be very clear. Um, but mission look something along the lines of this here. So to the left of zero should be above zero and to the right of zero should be below. And then likewise over here it should be below and then above, let me alone with her above, below and then above kind of like that. And on the other side, it should be above them below. Now that we have that, we want to estimate our intervals where the function is increasing or decreasing. And remember to do that we want to find where the function or wear a crime that is strictly larger than there are less than zero. So it's strictly large in the zero. We know the functional, the increasing. So to the left of zero, the function is positive. And then from 0.5 to 4, it's also possible. So we know it's increasing their and then, yeah, crime is less than zero. That tells us our country is going to be decreasing on the intervals. Well, it's going to be negative from 0 to 0.5 and then after four. No, just looking at what we have here. We can see that at each of these points, the derivative sign changes. So we know all these are going to be a max workmen. And we can say at X Z zero, we're going to have a so we're increasing and then decreasing so increasing and then decreasing. So this year is going to be a max at exit secret zero. And I mean local Max is not in global max. And then over here, where decreasing and then increasing. So this is going to be a local men and then at X is equal to four. We are going to be increasing and then increasing so that there should be a max. All right, now we can go in and look at the second derivative. You're out where functions can't give up. Come came down and all of that and again just to make it a little bit clearer. This should look something along the lines of so to the left of zero point by it should be negative and then positive. And over here around three, it should be positive. And then negative. All right? No, let's go ahead and by the intervals where it's gonna give up on computer. So for concave down our company, But let's start there. That means Double Prime Alexis. Be strictly locker zero. This is Com cable, and so it's positive to the right of 0.2 fun and then positive until we hit 3.46 and to figure out where it's going to be calm. Cave down. We just want to find we're after will from this lesson zero. And that would be the remaining control that we have a negative 30 2025 union, 3.4 sets to infinity, and we also want to figure out if he's air inflection points Will, since the sign changes all inevitable prime. That means each of our intercepts of F double time will also the inflection place. So 0.25 or about 0.25 there should be eight inflection point as well. A CZ at X is equal to about three. There should be a point of inflection, right? So now that we have these five points of interest, we could go ahead and put these into and doing that gives us this chart here. And so the reason why we want this chart. So we have an idea of what values we should go ahead and plug in. So these 1st 3 all came from information from crime and these next three er these next two all came from information of double price. So in my first graph here, I chose my ex interval to be from negative 1 to 6 because you might nose out of all these important points over, given our smallest value is going to be zero. So I just went ahead and show something slightly to the left of that. And then our largest value ends up being for so we at least wanted to be for to be from at least negative 1 to 5. And I chose to let it go to six because if we started out really zoomed out, we would see that we have this intercept here that we would still need to include. So just to push it out a little bit further. All right. And then for our y values. Charles Negative 202. 4200. Because here, the smallest value were listed at is negative. About negative 10 on our largest value is 4128. So I just went ahead and rounded the largest one up to the next 100 to get 4200 and then I chose it to be negative 200 because if I would have chose something like negative 10 it would have been really squished on the bottom there. So this here describes pretty much everything we want, and you can see that these points that I have are all the points that are listed on our graph here. So the only thing that we really don't know what's going on is in this little interval here. So what I did was I went ahead and blew this up to make it look like this. So just choosing those smaller X values, we would get a nice viewing window of negative 0.42 about winning and for the wine negative 10 to 1. And now we can see this and see all of the nice information we would get from that as well. So there's more than one nice viewing window that you can use prevents. But at least he's the ones that I think best describes the graph here.

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