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Produce graphs of $ f $ that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.

$ f(x) = \dfrac{1}{x^8} - \dfrac{2 X 10^8}{x^4} $

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we want to produce graphs of X is one over x l a minus two times 10 to the eighth Oliver. Excellent that we will all the important aspects of the curve and we want to then used it's graft to estimate intervals with a puncture increases decreases and levels of con cavity. And lastly will use calculus to find these intervals. Exactly. So I went ahead and just found what I thought was a good viewing window of this rap. And if you just keep on Zuma now, it looks like it just goes to negative infinity as Ex gets closer and closer zero because the value just keep getting larger. Margin margin. Okay, so let's just go ahead and find what we get for this. So the function will be increasing when so I don't want to talk about the dream that I just want to know about increasing. So increasing will be well. It doesn't look like it's ever increasing because it's getting really close to zero on the left and then it just keeps getting smaller, smaller direction. It is increasing to the right, so it looks like from zero to infinity, the function will be increasing, and it looks like it'll be decreasing from negative infinity to zero and then for con cavity. Well, it looks like it's just calm cave down on each side, so it just looks like it's going to be. Khan came down and it just looks like it's gonna be for negative. Infinity to zero Union 02 infant. So it doesn't really look like it's gonna be calm cape up anywhere. All right, Now, let's go ahead and use Ah, little bit of calculus, too. Find what these intervals action should be. Now. I went ahead and found the derivatives are ready for both of these functions. So the first since I could driven him. And now what we want to do is go ahead and set this function equal to zero. And so doing that, we're going to get that we really want to solve for when our numerator is equal to zero or 10 to the eighth times, X minus four minus one is equal to zero. And if we go ahead and solve that, that just gives us X is equal to plus or minus one over 100. So what this implies is we possibly have two points where it's going to be a maximum one minute. And now let's go ahead and solve for where? F Private exit strictly larger than zero f Prime of X is strictly larger than zero. Well, I went ahead and solved for this beforehand as well. Otherwise is gonna make the video way too long. So it's greater than zero on the interval. Negative one over 100 to 0. Union one over 100 two Kennedy No, until this year is increasing and is decreasing when F prime of X is strictly less than zero. And this was just the rest of our interval floor domain. Negative affinity to negative won over 100 Union 021 over. Now these two intervals air different from what? Our original graph. Let's so we obviously didn't have a good enough new ing window, at least for X values, because it seems for very small values. We have things going on, or I should say numbers with very small absolute about. All right, how we're gonna do the same thing over here. We're gonna set this function equal to zero, and that's going to say we need to figure out where 10 to the night times extra fourth minus 18 is a cruiser and solid. For that, you would get X is equal to plus or minus the four through of 18 over 10 of the month. And this here is approximately equal to was plus or minus 0.11 And now, using this, we could go ahead and soul for this inequality of double pie being strictly larger than zero or the function name Kong Cape up. And I went ahead and did that already giving us negative four through Oh, 18 over 10 to the night 20 union with zero for through of 18 over 10 to the night. And this function is going to be Khan came down, went double prime strictly Weston zero on the rest of our domains in trouble. So negative Infinity to negative four through of 18 over turn to buy union for through it too, turning to the line two then. So if we were actually used these critical values of these inflection points, we could actually see that a nice crop of this actually looks like this here. So it wasn't an issue of you didn't go up and down Enough. It was an issue of we weren't close enough to the Y access to actually see everything that was going on. So what this is they is sometimes these graphs that we originally have, like, here while if you were to just zoom straight out or to just zooms trading you would never really see what is going on because you just keep on doing out forever and ever and ever be released for a very long time. It just looks like it's going to negative infinity unless you really get far out. And by that time, that line will just look like a blunt. So you can't really see that is going to positive Infinity, actually, so this year is a nice photograph of it, and the viewing windows we would actually want to use would be between negative 0.5 to 0.5 and for wide negative 10 to the 6 to 10 to the third