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University of North Texas

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Problem 4 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) = \dfrac{x^4 - x^3 - 8}{x^2 - x - 6} $

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Video Transcript

we want to produce the graph of that rebuilds all of them for aspects of the curve, where it is X to the fourth minus X cubed minus a all over X squared minus X minus six. And we want to use the graphs of F prime in double time. To estimate the troublesome for the function is increasing, decreasing any extreme values. We may get intervals of con cavity and inflection points. So, uh, just for the sake of keeping the video as short as I can, I went ahead and already found F prime in F double time and already raft all three of these functions. So the first graf I have over here for FX that's just what it gave me when I plugged it into my graphing calculator. And I mean, that looks like a pretty nice craft. And you might, when you first put this in their sale. That's all I need and kind of be on your way. But if we use a little bit of calculus, we can see that this viewing window isn't the best one we can actually use. So let's go ahead and started looking at our first and second derivative graphs. So I went ahead and on my graphing calculator found the estimated the ex intercepts here. And these values will all be critical values that we're going to use and we can use the fact of where are function is greater than zero and less than zero. The driven it to Figaro, where the function is increasing and decreasing. So I just kind of do this off to the side here. So we know a function will be increasing if F prime of X is strictly larger than zero. So let's just go ahead and figure out what those intervals are going to be. Now these ah red dotted lines. I haven't both of the grafts or vertical ascent. Oops, because if you were to go ahead and factor the denominator, we should get um, X minus three and X plus two, meaning exited with a negative to an exit of the three will be vertical Assam tubes, and I just went ahead and put those in there to make sure we don't try to say that it's It's increasing from negative 2.745 all the way to negative 0.807 all right. So our interval of of increasing. So we first become positive. Like I was saying at negative 2.745 and actually like this meter. So at negative 2.745 and we're going to stay positive all the way to our vertical Assam toe, which is going to be a negative, too. Then we're going to union, and it seems that we are coming down also positive from our class until they're so it's gonna start from negative to again and then go all the way to negative zero point 807 Then we're going to be negative until 0.411 And then that will last until one point 82 And then we're gonna union that one last time with 4.5 nine to end Kennedy. All right, so that's pretty long interval. But at least we found it. And even better yet, we just got to estimate that, as opposed to having to solve it by hand, aren't now for words decreasing? We want f prime to be strictly less than zero. So to the left of our first point were negative. So, Megan, affinity to negative 2.745 Now union then would become negative again after negative 0.807 And we stay negative until 0.411 union with and then we're going to be negative from 1.8 to all the way into our vertical absoluto at negative or positive three. And then it looks like we start off it being negative on the other side of that brutal question goes well, and then we go all the way to 4.5. No. All right, now, let's go ahead and figure out if the's points are, um, Max is our bends as well. So normally, what we would need to do is to also check where our function is undefined. Our first and second derivatives are fine for critical values, but in this case it's undefined at X equals negative two and three, which are original. Function is also undefined. That so we don't need to worry about those being critical values. And we could just look at these five points that I have listed here. All right, so to the left of this point, here were negative or we're decreasing. So we're decreasing into the point, and an increasing after that means this is going to be a local minimum at X is equal to negative two points in the 45 then or other one were increasing and then decreasing. So that's going to say we have a local Max there. Our next point, we are decreasing because we're negative and then increasing after so that tells us, run out of men And over here we're increasing and then decreasing. So that tells us we have a max and for a last point where decreasing and then increasing. So so then we have a minimum. So we know whether these five points or Max is or men's. So now let's go ahead and look for the con cavity of our function. So at least these intervals won't be as bad. No, let's just go ahead. Over here, we know that a function will be con cave up when F double prime of X is strictly larger than zero. So let's go ahead and see where that is. Now it looks like to the left of our vertical ass. In total, we will always be positive. So from negative infinity to our brittle classes at Negative Union. And then we're positive again. Between negative 0.388 20.79 and then on the other side of our vertical ass in total were also possible. So from three to infinity now we want to go ahead in shack for where it's Kong cave down. And that's just going to be the rest of our interval or where after well, prime is strictly less than Sarah. So that's going to be starting from our first little Bassam tomb. So negative, too. And we're going to go until negative 0.388 union work and then, after a 0.79 all the way to our next vertical Assam tote being around three her at three. All right, so we have our own cavity now or are two ex intercepts here. Notice that they both change for being concave up or so Our first point goes from concave down to come. Keep up. So that means this year will be put an inflection, and then the other one goes from concave up to calm cape down. So this is also a point of inflection heart. So now that we have those points, those important points. I went ahead and found out what these values should be. So these bottom too, are our inflection points. And these top values will be all of our maxes or men's our local Max Instruments. So now the viewing window I chose, at least for the X value, was negative. Negative 45. And my rationale behind that is when I was looking, I saw that my largest value was about four. So I knew I wanted to be slightly to the right of that. And the smallest value I have is about negative three. So I knew I wanted to be the left of that. So one thing I did was I also kind of zoomed out. Really? For just to make sure I didn't miss any ex intercepts or anything like that. And luckily, uh, for us, I didn't. And so that's why I'm on that acceptable end for the why I just went ahead and found that my largest value was going to be about 30. So I just went a little bit larger than that. And then I just went below Negative. Uh, even though I don't have a negative value on my smallest one is about 1.2 here, or about 1.3. I just went negative, so we can also show these intercepts that we get a little bit nicer. So this is a really nice viewing window for this function and weaken. Clearly see that we have our vertical ask himto at ecstasy into negative, too, and not as good for exceeding the three. But it kind of looks like it. We see where we have our comm cavity, all of our Max's airmen's and all the little dots on this graph. I should mention our d local Max is men's or points of inflection, and we can see how it kind of changes using this viewing window pretty well.