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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) = \sqrt{x + 5\sin x} $, $ x \leqslant 20 $

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we want to use some kind of cat software. Two. Graft. The function of X is equal to the square root of five time Cy in its prospects, when X is less than 20. So I went ahead and did this here. So the first thing we might want to do is you just write down where the intervals are, because later on, we're going to want to find the person second redness and determined intervals, where the functions increasing, decreasing as well. A cz notables of cock cavity. And if we already know what our domain is, it'll just make making those intervals a little bit easier. So let's go ahead and do that really quickly. So this first animal here is going to be about negative 4.96 to about negative 4.15 Then our next interval is going to be 0 to 1 point for 105 And then it'll be about 496 2 20 and who we actually look at the person second derivatives. You might notice that we have a bunch of points that we should possibly see for our first relatives and in these 1st 2 a little, he says. It doesn't really look like calm cavity changes. But then maybe around these points here, it looks like we have changing con cavity. So just some points we might wantto look out for when we're looking for are Venables of contact ity and where the punctures increasing decreasing. I would have had add plugged that in and found f fun to the five X Plus one Oliver two times the square root of five X plus X, and I went ahead and found where these intercepts work, and now we'll do is let me go ahead and screw this up about that more brutal. Let's grow if he's going to the max is your amends, and at the same time we could kind of go over What are intervals of increasing decrease or going to be so first? Remember F prime of X rated strictly greater than zero. Then we know the function is increasing and it f prime of X is strictly less than zero. Then we know the function is increasing, so to the left of negative four points 511 it's positive. So I know it's increasing into this point and it's going to be decreasing afterwards, so we know this is going to be a local Max are other Intersect herbal. It's positive toe look, so it's increasing into it. And then it's negative after. So it's decreasing after. So we get a max there, our next intercept again, it's increasing into it has been increasing Gerstle of another Max. And now our next point well, about this one's gonna be a minimum because it's decreasing into the point and then increasing afterward. So men are other point Oh, it's increasing into it has been decreasing out. So this is going to be a max and there are other function. Are other intercept here? Well, it is decreasing toe increasing to the right, so that means we have a minimum here. Now that we have that, we can just go ahead and write down all of our animals, as we already said, where it's increasing, decreasing So from negative 4.962 negative 4.511 and then from 0 to 1.72 and then from 4.9462 0.55 and then 10.742 14.379 and then our last interval works increasing will be 17.772 kind of long in trouble. But luckily for us, the decreasing one won't be as long. Actually, yes, it will be just a long unfortunately. So it starts to be negative after negative 4.511 And it will stay that way until negative for 0.1 so far and then union with and then 1.722 up until about 4.15 union. And then after this 8.5 by 2 10.79 or and then our last winter Bowl, where the country's decreasing is about 10 points 379 to 17.7. So we found our maximum men's as well as with. Function is increasing and decreasing. And if we just kind of go back and look at these values, you see that they match up with all these little lines that I have drawn on this road. So it would be good to know our intuition gave us some of the answers. No, I want to have found the second derivative and then grafted as well and found the intercepts and just the weekend look. So to the left of 9.62 the function is gonna be calm. Keep down. So let me just do it like this. Remember, when F double prime of X is strictly greater than zero, it is one to be calm. Keep up. And when F double pride of X is strictly less than zero, the function is going to be calm. Cave death. All right, so now let's just first go out and find our points of reflection. And then after that weekend, right are intervals where it will be called gave up the concrete down. So at nine points except you well, to the left of that, the bank is going to be calm. Keep down into the right. It's gonna be concubine. So that is a point of inflection to the left of Well, it should be conquered up into the right. Kong Kate down since is also going to be appointed inflection around 15.8. It is concrete down toe conch ate up the rate So we have another pointed inflection and our last one will be concave upto left a concave down to the right so that will also be a point of inflection. Now let's go ahead and write our intervals. So we're Conkey up when F prime of X is larger than zero. So that starts at 9.62 and it'll go in hard. Next. There's 12 0.252 I've been with Union that with 15.811 until 18.651 and I will be all where functions come. People now is calm. Keep down. So is concave down on these two animals here, which are going to be negative? 4.962 negative for 0.105 you him at the zero to 4.105 and then from 4.96 to our first X intercept being 962 and then it's negative again around 12.22 until 15 811 and then our last interval will be 18.6512 20 are in front. And so we were to look for those inflection points on the original ground. Those four match up with what we said and we can also see those 1st 2 A little hoops are going to be was concrete down. And then after that, it just kind of follows that interval pattern that we have. So we found everything we wanted for this problem.