Problem 1 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) = x^5 - 5x^4 - x^3 + 28x^2 - 2x $

Answer

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Video Transcript

so we want to produce The graph of f of X is equal to X elective my sexy for my ex excuse plus 20 expert minus six and in particularly we want to use the graphs of F f double time to estimate our intervals where the focus is decreasing and increasing any extreme values. Intervals of one cavity and inflection points. All right, so just to kind of saved time. So in this 1st 1 here, I went ahead and already plotted the graphs of F f prime and double print. And remember to find these derivatives, we would just go ahead and use power will. Now, in this first raft over here, when I just plug this into my graphing calculator to start, you can see that the the viewing rectangle that it gives isn't all that good. So now if we use a little bit of calculus, we can figure out a better viewing practical for it. So when I take the first derivative, if this were interested in where the function is increasing and decreasing, all we really care about is where all the intercepts are. So I zoomed out far enough so I can see all of the intercepts. And since this is 1/4 degree polynomial and it has 1234 intercepts here, we know this is all we really care about. So what we cannot do is say that points that may be of interest are going to be these intercepts because those are going to be our possible critical values. And we can see that we can't estimate our intervals of increasing and decreasing by just looking for where this is positive, where it's negative. So we know f Prime Becks is going to be strictly larger than zero on. So to the left of negative 1.5 So negative infinity to negative 1.5 and then after 0.36 to 2.6 to three. So who point? Next time I go ahead and write sold it over here 2.62 first, 0.36 to 2.6 to three, and then our last Interpol, where Obi larger will be 2.841 to infinity. I'm gonna scoop this up a little bit now we can go ahead and find where its lessons there, which would just be the rest of our intervals. So, uh, prime of X, strictly less than zero. So, between negative 1.5 and 0.36 it'll be negative negative one point 20.36 as well as 2.62322 point 841 All right, so if we can see that all of these have a change in con cavity in a a change in its derivative. So to the left of negative 1.5, it's positive to the right, it's negative. So that means at 1.5 there's going to be a well if I increase into something in the decrease after by the first year of the test, that tells me it's going to be a max. At 0.36 we're going to have a men that occurs at 2.683 word decrease There were increasing into the point of in decreasing after, so this is going to be a max, and then the last one is going to be a minute. All right, so we at least know when we're looking at are viewing rectangle. We should at least go from negative 1.5 to 2.84 out of minimum, because these have important X values for us. Now let's go ahead and look at F double prime. So here this is third degree polynomial. And here we have our three intercepts. So we at least know this is as wide as we need to do it. And just like before, we're going to look for where it's larger than zero lessons. Areas will be confined. Conclave of income came down and maybe I should say appear. Remember, F double prime for F crime larger than zero is a increasing in trouble, and F prime less than zero is a decreasing in trouble. Now let's go ahead and figure out where F double prime is listens your own grave. So when a double prime it's strictly larger than zero, this tells us we're gonna be conking out. So our first interval of that is going to be from negative 0.888 to 1.153 and then we'll be positive again. 2.735 to infinity. Now we can go ahead and find our function is calm. Kate down or when a double prime. It's strictly less than zero, and I will just be our remaining intervals negative and 32 negative 0.8. And we also have 1.15322 point 73 Let's go ahead and see what's going on with these points. So we're changing con cavity at each of them. So to the left of X is equal to negative 0.8. It's down and then up. So this is an inflection point. Likewise, for 1.153 it goes from up to down and then two point 735 of those from down to up. So we know all three of these will be points of inflection. All right, so now we have some ideas of X values that we need to include in tor craft at a minimum. So I went ahead and plugged these values into a graph here and or into a into the equation after Becks. And so the reason I chose my interval for the X value soaring from 9 to 3 to three is my smallest value. Four. My ex is is negative 1.5, and just to make sure we get the rest of our in behavior. If we were to just have negative one point by which would be, um this maximum right here. We wouldn't include this zero. So when you would first graph it the first time you see that you're still missing some in behavior. So then you just go ahead and go a little bit further so we could get our other X intercept in there and I chose the interval three since our largest value on X was about 2.7. And when we would grab that, we would see it's going up and still increasing, so I would just know the end behavior would be going to infinity. So that's why I just chose three and didn't really have to go much further and then likewise, for the why values that I chose Well, my smallest value waas this one here. So about negative 0.3 So I just shows negative one. And for my largest value, I went ahead and chose 50 for 60 because my largest output that I get for these important points is at 56 about Okay, so that's why I chose my viewing window, but you might notice that at our smallest value here, it says we should have something going on. But if we look at that little in trouble right here, just looking at it on this graph, it still doesn't look very good. So if we go ahead and blow that up, we can actually see that around that point. We do actually have a local minimum. So this is how you would go about solving these kinds of problems. To get a good viewing window of your craft, you would find your points where you have Max's men's or just critical values as well as possible points of inflection, and then play around with your new window until you get everything in there. And then if you have any kind of weird behavior, kind of like we did around ecstasy with zero, you could just blow those up to go ahead and give a better idea of what it should look like. They're