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) = 6\sin x - x^2 $, $ - 5 \leqslant x \leqslant 3 $
we want to produce graphs of f of X equal to six times sine X minus x weird on the interval Negative five, 23 That renews all the important aspects of her, and in particular, we want to look at the drafts of a crime and a double prime to estimate the intervals where the functions increasing, increasing in the extreme values, intervals of con cavity and points of inflection. So you might notice I have these three graphs here. So the first graph is just what I got when I plugged this into my calculator on Interval Negative 5 to 3. So it looks like a captures the essence of everything. But maybe we use calculus. We might find that some of the points or some of the behaviour is kind of hidden with just how the shape of the graph It's all right. So let's go ahead and first start by finding our place of increased n decrease our intervals of increase and decrease. So remember that is going to be where F Gilbert F Prime Prime fix is strictly larger than zero. This tells us where the function is increasing now. Maybe I should give a little bit of intuition as to why I have them crop like this. So all I did was make sure that or the derivative of the second derivative that I captured all the intercepts that the RAF had because anything we care about these function will be just revolving around. Our critical values are inflection points were which are all going to be exact. So I really don't care about the overall structure of it to find where it's increasing, decreasing, uh, anything else of importance. So go ahead and use this now to find so it looks like to the left of negative 2.938 The function is positive. So will be increasing their and we know we need to stop that negative five since we're not going to infinity in this case. So our first interval of increase will be negative. Five two negative, too, point 938 Then after that, it starts increasing again at negative 2.663 and it keeps on going all the way until 1.17 So that's going to be increasing. And then decreasing is one. F prime is strictly less than zero and this would just be the remaining in trouble of our domain. So the function is going to be negative from negative 2.938 up until 2.663 And then we're gonna union. That would Well, it becomes negative. Begin after 1.17 and this is going to stay. And I get all the way until our other boundary or in point being three now. Well, we could go ahead and do is check to see if these air Maxie Germans so starting all way on the left to the left of that F prime is positive. So it's increasing into this point and there's going to be decreasing afterwards. That tells us we're going to have a local max here. Well, then, or other one where decreasing into it and then increasing after. So this is going to be a local local blacks, local men and here on the other side, well, we're still increasing into the point and then decreasing after so this year will be a local max. All right, now we could go ahead and look for con cavity, our function. So remember this is going to be Kong cape up when at double Prime is strictly longer than zero. And we can see that that occurs between negative 2.82 up until negative zero point three. And then the rest of this will be when f double crime is negative. We're strictly less than zero, which would tell us, where is Kong cave down. So it's going to be from negative five two negative 2.802 union with negative 0.382 three. And here we start concave down to lift. And then after this first point will be conk it up. So this will be appointed inflection and our other value is terms Khan gave up and then goes to calm cave down. So this other point will also be a point of inflection. All right, now, I went ahead and took those five values that we got from the last pardon. So these 1st 3 are all of our local Max men's, and these other ones are inflection points. And I use these to figure out what my scale for Why should be so The ex was already determined. It should have been negative 5 to 3, so I didn't really need to play Rama, but and I saw that my largest value was here at about 4.1. So I just needed to make sure I was a little bit higher than that at five. And I chose my bottom. We knew to be negative 20 even though the smallest body that I end up getting produced is about negative 9.9 because I also wanted to check for by end points to make sure that neither of those ended up being a maxim or men or and it just happens to be that atnegative high. We end up with a global men. So I just went ahead and went out a little bit more or the interval. Now this looks pretty much I mean, it is exactly the same graph that we had before, but if you look at all these old dots, these air, those five points I have over here on the left in my chart and we have this cluster of three right here and if you were to just look at that, you might overlook that in the original draft and just say, Oh, yeah, that would just be a point of inflection. And you would never possibly guess that there might be a local men in the local max. And if we blow this up, which is this crap over here, we can see that, Indeed, this here is a men. This one here is a max, and that middle point is a point of inflection. But like I was saying, if we just looked at that first graph, I mean it, you little bit of calculus. We would have never saw that. Every blew up that little rectangle right there. We had some extra pieces of information that we were actually missing.