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Describe how the graph of $ f $ varies as $ c $ varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when $ c $ changes. You should also identify any transitional values of $ c $ at which basic shape of the curve changes.

$ f(x) = \ln(x^2 + c) $

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We want to describe how the craft of F of X is it through the natural log of X squared, plus C Berries at sea Berries. Then we want to graft several members of the family to illustrate the strings that were discovered and in particular will want to investigate how the bathroom of minimum and points of election move when see changes. We could also identify any time transitional values where the basic shape of the curve changers as well. So first, I'll just go ahead and figure out for what values of X dependent on see this function will be defined for. So we know that for our domain, well, we can't plug in zero or negative values, So we're gonna want X squared plus C to be strictly greater than zero practice. See over, I guess X squared sugary crazier than negative C uh, and then we can go ahead and take the square root on each side so they'll give us two different solution. So X is going to be strictly greater than the square root of negative. C and X will be strictly less than the negative square root of negative seat. So one thing that we will need toe keep in mind is so it's C is strictly greater than zero. And we can also just kind of conclude this from this line here. Then there no problems. So that means we'll have all real numbers for coming. So let me just go ahead and say so all roll numbers or don't make. But if C is actually less than zero, then this here will be our domain that we're working on. All right, Uh, let's go ahead and find some ex intercepts also. So necks intercept, so we need to set that equals zero. So natural log of X squared plus C equals zero, but we need a exponentially it on each side so they get X squared. Plus C is one. Subtract the sea over and square root so we get X is equal to plus or minus one minus seat, plus or minus the square root of one minus e. And so, just like before, Well, if we have the case of one minus C, being lessons are well for us to have an X intercept. I should say we'll need one minus C to be greater than zero or we'll need C to be less than one. So we only have X intercepts given, huh? C is less than one. So at least from these two things, we might start to think that we might have some transitional values around. C is equal to zero and season. So maybe C is equal to 01 transitional. So those might be some graphs we might want to look at. Let me just put a question mark. Now let's move on to the derivatives. So f prime of X. So they take the derivative of bachelor log. Remember, it's one over whether we have on the inside. So it's gonna be one over X squared plus, c. Then we multiply this here by the derivative on the inside I general to acts of disability to exp Ober X squared plus C. And so we go ahead and set this here equal to zero. Well, that tells us that X equal zero. And so this year does not depend on what we get for our value of see, So this will be a maximum or minimum X is equal zero. But we have to keep in mind that what we found earlier for the name. If C is less than zero, we won't get this point. So this is only for one see is strictly larger than zero that we will get this maximum or minimum point. All right with them. Let's look at the second derivative, so to take second driven and we're going to use questionable. So be low. The highs of the derivative of two X is too and in minus. Then the opposite order so high, do you Lo and the derivative of two X squared plus C is going to be two X and then all over cabin the denominator square, so we could go ahead and simplify its here. So this is going to be or expired or Maya sparks where and combining those together we will end up with negative two x squared plus two see all over X squared plus C squared. And now, if we were to set, this here equals zero. Well, we only are interested in our numerous equal zero. So that's going to sit two x squared, is it to see two by twos? Get exited to plus or minus the square root of see, So this will be our possible point, uh, inflection. I'm so remember, though this will only be when C is strictly greater than zero, since we cannot take the square root of a negative number. So let's go ahead and see if this compares to what would find all right, so we can go ahead and see that. So starting with season with zero, well, it looks like the only point that we are not to find out is at Caesar's zero, and our ex intercepts look about where we would expect them to be when Season one we see that our domain starts to lose a little bit more and it pushes the X intercept slightly further out as well. A cz were C zero to negative vibe that pushes them slightly more out. And we don't see any Maxtor men's or points of inflection for that. Then s so we thought that sea is zero might be a transitional point. So we said this might be transitional as well. A season or 21 he said, might be transitional. So going from seizing with zero to cuz consisted of five. Well, we can see now that the function one becomes continuous. So this definitely was a transitional value around cuz zero and then at X is equal zero. And for all of these, we see we end up getting aid minimum about so that part, their checks out and then weaken, See around maybe here on each side. And maybe it's further out over here and it looks like it's going from Kong cape up concrete down to come pick it up and then to conchita again. So we end up having our points of inflection, aligning as well with what we found before. So maybe it's easy to the one you can call it a transitional, Um, because of the ex intercepts. But for the most part, maybe this really isn't one since it the shape of the graph doesn't really change all that much. The only real difference between scenes even deserve 10.5. This easy with five is it moves further so the minimum becomes larger. But actual shape still kind of has. That same bill is just not as exaggerated. So I would say V six craps here pretty much describes his family pretty well as well as everything we found on that first page