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# Use the guidelines of this section to sketch the curve.$y = \sqrt{x^3 + 1}$

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the domain for this one is exes. All riel for intercepts. If we set um X zero, then why will be one? And if we set, why does zero than X will be negative one? There is no symmetry for this graph. Um, ask himto Sissons kind of tricky. So let's take each one to the power of three to clear the 1/3 here. So here, if we see X becoming very large than the one is negligible. So we have and then we take the square root. So let's see this. It's negligible. So why cubed is X cubed approximately, Let's say approximately and then take the cube root. So why is approximately ex? So I want to say that there is an oblique ass in tow near why equals X? The next thing we have to do is find why prime to do the first derivative test two shores increasing in decreasing. So why prime is 1/3 execute close one to the power of negative 2/3 times the derivative of the inside, which is three x squared. So why prime equals X squared times x cubed plus one to the power of negative to over three. And let's set that equal to zero to find the critical points. So our first critical point is zero, and our second critical point is negative one. So if we do the first derivative test, this is a prime. Now we're testing values around our critical points. So we have negative one here and zero here. Anything smaller than negative one will give us positive. Anything in between. Negative one and zero positive and greater than zero is also positive. So I'm gonna call negative one and zero both saddle points, subtle points. Okay, so that's it for the first derivative test. And now we're gonna take a look at Kong Cavity. So we need the second derivative for that. Why? Double prime is gonna be two x over X cubed plus one. And that's all to the power of five over three. Set that equal to zero, so x zero. And if we take a look around zero for our second derivative F double prime, it's positive before zero and after zero. So they're con cave up for both. So we know now that it's the graph is always increasing, and it's Kong cave up everywhere, and we have saddle points at negative one and zero. So let's try to grab that intercept that 01 and negative 10 Let's say this is one. This is native one, and now we know it's always increasing in con cave up, so it's gonna look something like increasing con cave up. And here we have a saddle point and here we have a saddle point, but it's always gonna be Khan gave up. It's always gonna be increasing. So it has to look something like this in order for it to be always increasing. And keep in mind that there is a vertical ass in tow at approximately y equals X. I mean ah, oblique ass in tow. Sorry. So that our graph approaches but never touches this line.

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