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University of North Texas

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Problem 17 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{x^3 + 5x^2 + 1}{x^4 + x^3 - x^2 + 2} $

Answer

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Video Transcript

we want to use some kind of cast software to map F of X is equal to x cubed plus five X square plus one all over extra foreclose excuse minus expert posture. And we want to also use that same software to buying the first and second derivatives and then used the grafts from our derivatives to estimate interval of where the functions increasing, decreasing con cavity as rolling by any inflection points or extreme bounds. So I went ahead and plugged in, uh, just into our graphing calculator, and I kind of zoomed out far enough and did a little bit of looking to see Where are there any ex intercepts and all of that to kind of get a good viewing of our graft? So this is what I decided for that viewing window. Now I went ahead and used that same software to take the derivative, and you can see here that this thing is kind of bulky. So glad we were allowed to do this using a graphing Maria cast software as opposed to doing that by hand. And just like before, I zoomed out far enough and then looked for any kind of areas that I think might have been of interest so I can ensure I get those interviewees. So, um, just starting from left to right. So in that little square there, if we zoom in, we can see that to the left of this were negative. So where Decreasing into about negative 9.5 and an increasing after. So we know that about exiting to 9.5. We're going to have a local men using the first relative test at our next one is over here at about negative 1.294 and we can see we're increasing into that point and then decreasing after. So we know that this year should be a local max. Our next value is going to be at X, is equal to zero, and we can see that we should be decreasing into that point and then increasing after. So this here would be a not max, but a local men and then for our last one. This is about at 1.55 and we are increasing into the point and decreasing after so we know we're going to have a local max at about X is equal to 1.55 Now, let's go ahead and actually write down these intervals. So we said that when F prime of X is larger than zero, the function is going to be increasing. So we first become positive after negative 9.5. So negative 9.5 and then we stay positive until we hit our next intercept at negative 1.294 And then we become positive again at zero until we have our next intercept, which is 1.5 now to find where it is decreasing, we're going to look for where f Prime of X is strictly less than zero. So that is to the left of our first nurses so negative there, too negative time point, but union with than negative 1.294 20 and then union at 1.552 in Bennett. So we found our local max is local men's as well as our intervals of increasing decrease. So now we can go ahead and go to the second derivative and again already went ahead and used our caste software to find what the derivative is going to be and again, I am thankful that they don't want us to actually have toe get that my head, because I would hate to have to take those driven. It's and simple. So just like before, I just went ahead and assumed pretty far out and then looked for some behaviour that might be interested along the extremities. And this is what I think would be the best feeling indifference. So just starting from the left. So I went ahead and blew up around negative 14 because of round there Way. See, we will have a point of interest. So this right here is about negative 13 0.806 and we can see that we are Kong Kate down. Remember, the second derivative will tell us Con cavity. So is calm. Keep down into it and they coming out will be concubine. So that tells us a round here. There will be appointed inflection for our next value. We have negative 1.55 about, and so this here will be Kong cave up to the left and then Kong cave down. So this year will also be appointed inflection, and then our next value is about negative 1.2 So it's Kong Cade down and then calm cave up. So that's also a point of inflection. Our next intercept is 0.602 and it is Kong cave up to look Kong Kate down to the right. So that means we're gonna have a point of infection there. And then for our last point right there, that's about one point or 87 and is going to be Khan came down to luck and conclave up. So that means we also have appointed infection. So about around all of these expertise here, we should have a inflection point. Now, let's go ahead and actually write out these intervals. So, like I was saying earlier, it will be Khan came up when F double prime of X is strictly larger than I do. So that looks like it is about negative 13.806 until we hit our next intercept, which was negative point or negative 1.5, and the union that with negative one point region until we had 0.602 and then we're positive again at 1.87 or 1.487 And then that goes off to infinity. And now for concrete down. It'll just be rest of our interval or when double fine strictly lessons there. So just don't be from negative. 30 two negative. 13.86 union with negative 1.55 to negative 1.2 and then union that with our last interval being from 0.602 to 1.487 So we went ahead and found where our functions going. We can't give up concave down over inflection points. So this was the last step we needed to do for this problem.