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Use the guidelines of this section to sketch the …

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Problem 1 Medium Difficulty

Use the guidelines of this section to sketch the curve.
$$
y=x^{3}+3 x^{2}
$$

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Video by Mary Wakumoto

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Video Transcript

Okay here, we have a function Y equals execute plus three X squared. And we want to graph it, we know because it's cubic, it probably has some general shape like this. Um But let's go ahead and figure it out more. Specifically, I'm gonna go ahead and find the zeros first. I'm going to notice that experts in common leaving behind X plus three. So we know that we have zeros whenever X equals zero or Um X equals -3. So I'll go ahead and do those right away and that will give us some help with the graphing. And I can certainly just keep plotting points and I would be able to get the general shape of the graph. But let's use some derivatives. So let's keep going and find the first derivative. First derivative will be by power roll three X squared plus six X. I can factor out a three X. Giving me X plus two. And if I set that equal to zero, we find out that we have critical points at X equals zero And x equals -2. And so I'm going to go ahead and make a sign chart and keep it running so I can see what's happening. Okay, so at zero and at uh whoops, let's put the zero on the right since we need room for the -2. Okay, so we will put zero here minus two here, that's our X and our f prime of X is zero at both of these locations. Let's go ahead and plug in values to find out whether f prime of X is positive or negative in the different intervals if I plug in something bigger than zero, like one, I'll definitely get a positive. If I plug in -1 I will get a negative. If I plug in -3 I will get double negative. So that's a positive. Alright good. So basically I'm increasing decreasing, increasing. Alright now let's take a look at our next derivative. Let's do y double prime. That'll be six X plus six by power role. So pull out the six and I get X plus one And that equals zero when x equals -1. So let's plot that on our chart -1. That's the zero for f double prime. And let's again do the signs when you're bigger than -1, you definitely get a positive. Um So you get a positive positive and when you're um less than negative one then you get negative. And then this is how I can I really love doing the sign chart because this is how I can no the kind of general shapes. So I have Less than -2. I'm increasing concave down. So I've got this shape And between -2 and -1, I'm decreasing because I have a negative first derivative but I'm concave down. So decreasing concave down. So that would be like this Then -1 is zero, I'm decreasing. Um concave up and then at the last bit I'm increasing concave up so I get my general shapes which is cool and alright so let's find also like max and men we know max occurs when F prime goes from plus to minus, so this will be a max point and um this will be an end point. Um and then here's a point of inflection because that's when F double prime changes sign point of inflection. Okay, so let's get the actual points. The max occurs At -2, let's plug it back into the original. So it's four times one. So four, the min occurs at X equals zero. So it was like that and we get zero. Good to know that are a point of inflection occurs at -1. Let's plug that in. I think it's this 1 -3, so -2. So now we have um all sorts of points up, like and we double check, I did that right, If I plug in -1 I get -1 a plus three. So that's actually positive two. I was looking at the wrong thing. Okay so let's plug in those points and we'll be able to finish drawing our graph. Okay, so -2 for we have a point up here, We have a point at my a point of inflection and -12. So that's the point of inflection. So basically we know that we have this shape. Um then it um switches to concave up because we have so far we've done that shape and that shape. Now we're concave up decreasing and then we're going to be concave up increasing as we get past the men. So let's just plug in one point on the right side. How about plugging in one? We plug in one, we get four. So then that would be up here and we have I think that's sufficient. We have drawn our graph, no horizontal as some toads, no vertical ascent. Oh, it's just a nice wonderful cubic function. So anyway, that's a lot of fun. Hopefully it helped have a great day.

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