Use the guidelines of this section to sketch the curve.
$ y = \ln(1 + x^3) $
All right. So the domain for this one is X has to be greater than negative one, so we can keep our argument positive. And for intercepts, we have 00 So the origin symmetry, we have none. No symmetry. Um, assim toast. We have, um, one X equals negative one. All right, because our domain cannot be, uh, negative numbers because of the natural log. So that's why we haven't asked into it there. All right. And next up, we have to find increasing and decreasing intervals. So why prime my prime equals three x squared over one plus x cubed set that equal to zero. We have X equals zero is one of our critical points. All right, so first derivative test. So this is a crime. It's test points around zero are critical point. And we see that it's gonna be increasing always. Okay, So our function is always increasing. And if we find our second derivative now for the con cavity tests, why double prime equals three times to x minus x to the fourth on all over, um, one plus one plus x cubed. And that's all squared. Set that equal to zero. And we come to to, um, critical points X zero and X equals the cube root of two, which is approximately 1.3. Okay, so second derivative test. So it's double prime. So this is zero. This is, um, the cube root of two. We have, um, conk it down. Oh, my God. That's come concave down here. Con cave up here and in Kong Cave down again. Okay, so these air both inflection points. And, um, just so we know the why value so f of the cube root of two. The y value is gonna be, um, ln of three. So that's approximately 1.1. All right. And then it was just so we can graph it, leader. Let's keep all of that over here. And now we can graft. So we know we haven't asked him to put that in in green. It's gonna be a negative one. So this is negative one. We have a vertical ass. Until here, we haven't intercepted 00 We have, um, it's gonna be Kong caves down until it reaches zero. So Kong cave down until it reaches zero, and it's always gonna be increasing. And then once it reaches zero is gonna be con cave up until it hits our inflection point. Um, so let's say this is, um 1.3. So this is if this is 11.3, be about here and then if this is one 1.1 1.3, that's it. Right there. So that's our other inflection point. So it's gonna be con cave, um, up until it hits her inflection point, and then it will be conquered down the rest of the way up.