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^6 - 5x^5 + 25x^3 - 6x^2 - 48x $
we want to produce a graph of is equal to extra six minus five X to the fifth plus 25 Execute minus six X squared minus 48 X and want to make sure that we include, um, intervals of increase, decrease extreme values, intervals of con cavity and inflection. And we're going to use the first and second derivative grafts to do this as well. So this first graph here on the left is just a f of X assumes I plugged it into my calculator. Eso doesn't really tell us much about the graphical, unfortunately, but with a little bit of help from our first and second ribbons, we might be able to get a good window to view everything. So I went ahead and took the first derivative of this implanted it as well, and I just pretty much expanded it out until I had saw all the intercepts. And so here we have 12345 intercepts. So this is a 50 50 degree polynomial. We know that satisfies everything that we need now. I went ahead and labeled these points here because they also want us to find where the function is increasing and decreasing. So remember, when F prime is strictly larger than zero, the function will be increasing. And now we can go ahead and say that. So, uh, we know it. Zoom out a little bit, get more space. So the function is increasing, or f prime of X is strictly larger than zero on. So would be between negative 1.307 two negative 0.839 And let me just actually go ahead and get this all the way over and the union that with. So then we become positive at X is a little one, some 1.6 two, then two 0.505 and then union that with our interval after 2.7 for eight to. So that is where our anxious increasing and we will know the function is decreasing on all the other intervals or when f double proper F prime is strictly lessons. So that would be to the left of our first point some negative infinity to negative 1.307 Union Negative 0.839 to 1.6 Union two point 505 22.7 or eight. So now we look at this, so it's our first point here. Well, it's decreasing, so it's decreasing into that point, the increasing after. So this here is going to be a minimum at the next one. It's increasing and then decreasing after. So this year is going to be a max local max and following the same thing. The next one is going to give us a minimum, and then we're going to get a maximum and then a minimum after that. And remember, that's just by using the first derivative test. Now, let's go ahead and look at our second derivative here. And so just like before to get that viewing window, all I did was Uma until I saw I had four routes because all I really care about is where the functions positive it, Nick. All right, so this here is going to be Kong Kate up when it is positive. So oh, is when f double prime of X, strictly larger than zero. So to the left of negative 1.1, it is positive. So that's negative. 32 negative 1.1 union and then 0.0 two. You're a 10 feet to 1.718 and I'll go ahead and just put this underneath. Union 2.635 to Infinity, and the function is going to be con cave down when F double prime is strictly not greater but strictly less than zero. So that's going to be just the remainder of our interval. So negative 1.1 two a 1.120 point 08 union 1.718 22.63 All right and or points of inflection. All we need to check to see is that it's possible one sided, negative on the other. So so our first point here that does change cavity here. We also changed cavity changing con cavity as well as change cavity. Just because we want to see where it's positive ending where it's negative. And if it changes from positive to negative, negative positive, then we know we will have a point of inflection there heart Now, the next thing. I went ahead and did it. Waas plotted all those X values or all the ex intercepts for the person second derivatives because we know there's air going to be important points. So the's first handful are all either. Max is Airmen's local Max. Is there a men's, I should say, And then the rest of these are our points of inflection. So I went ahead and planted all these, and I looked at our output out use. So here you can see the window I ended up choosing, and the reason I chose this window is I looked at our X values. So my smallest X value ended up being negative 1.3 of seven, and our largest ended up being 2.7 or eight. So I at least wanted to get these values into there. So I just slightly chose something to the left of negative three to the left of negative 1.3 just to make sure I got everything else in there. And then I let it go to four, even though it was just 22.7, because I also wanted to include this intercept here as well, and the Y values Well, I just went to all the outputs of our important points, and I found the smallest being about negative 33. So I just went ahead and went slightly below that negative 40. And then I look for our largest value, which occurred at about 22 are actually 23. Up here was our largest value, and I just went slightly above that one as well. So this year, I'd say, gets all the points of importance and the little dots you see on our graph were all the X values. So either that Max is airmen's or points of inflection, and so we can definitely see all the changes. Whether it's changed a cavity maximum, I would say this is a good viewing window for this.