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University of North Texas



Problem 31 Hard Difficulty

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) = e^x + ce^{-x} $


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Video Transcript

We want to describe how the ground L f of X is equal to X plus C times negative active areas of sea Berries. Then we want to grab several members of this family to illustrate the strains that we discover and in particular, we want to investigate how the maximum minimum and points of inflection move when see changes. He also should identify any kind of transitional values where the basic shape of the curve changes also. So, um, first, let's just go ahead and maybe find what are ex intercepts. Should be that might be something nice to know. So to do that, we could go ahead and set This equals zero. But before we do that, I'm going to factor out e to the negative X I get e to the negative X. Also, it's gonna be e to the two X plus C. Now, if we set this here equal to zero, well, we don't eat a negative X, remember, be zero since each of the negative X is always positive. So that's gonna tell us e to the negative. Two X has to be negative. C take the natural log on each side and that'll give us two X is equal to the natural of negative C and then divide side by to get extra between one. How natural of negatives now something you might notice is that this will only be defined when C is strictly less than zero. Since we can't take the natural log of zero more, can we take the natural log of the baby so we only have X intercepts? One. See a strict lessons really looks like what now? This year's f of X? Let's go ahead and find our first derivative so we can possibly find r maxes amends with the driven of Evita vexatious vex and the derivative of E to the negative. X is just negative be tonight vex ill t minus c e to the negative x So once again, less factor out that eat negative x So even that little extra e to two x my seat And again, we're gonna set This here equals zero. Uh, it's a negative x camp zero. So the other part has to be zero so e to the two X is a c natural log of capeside. So that would give us two exes it to the natural log of see divide by two, we get X is equal to 1/2 natural log of C and now we will only have where this is a maximum or a minimum when X is strictly are not experts. See, it's strictly larger zero for the same nationality core. We can't take the natural log of something that zero, nor come and take the natural order something that all right, and let's just see real quick, and this will be a maximum or a minimum value. Actually, we can see that when we actually start to graft, right? So then, for the second derivative, cool, taking the second derivative of this will be e to the X and then again we'll just change the sign. So we plus c e to the negative X. So if we said this year equals zero, well, this is the same thing as F of X, so we already know right away that X is going to be 1/2 natural log of negative C So see Haas to be strictly less than but But this only occurs when C is strictly lessons. So this would be our inflection point. IFC is less than zero. This will be our maximum or minimum when she's great and zero and this will be our X. All right, so let's go ahead and see if this matches direction before we do that. Uh, let's do one other thing. So know this If we were to just like C equals zero epa, Becks becomes eat. That's and we know the derivative Eat, Alexa, just eat of X. So this year would have no maximums nor minimums. But we might say that around season going zero, this is probably our transition that they were saying we should Look, we're just because each of exes will have different properties use then when we're adding to the negative X. All right, So I'm gonna head and graft a couple of these here, and, uh, let's just go ahead and write. What are maxim in? Should be so remember our max or men. And in this case, we can see that it should be minimums because we said that we only got a minimum. Where are Max? One scene was strictly larger than zero. Remember? That was gonna be at X is equal to the natural log. Oh, 1/2. See and we know our point of inflection was only one c was stricken. Listen, zero that was X is equal to 1/2 natural negative. See? And remember, this was also the same is our ex interesting? So at C 0 to 0, we see we just get the function of X. And then when we start to go towards the positive values, we see that so for seeds he could wind well, natural. Long of one is zero. So then our minimum would just be an extra zero and been as X increases. So does where the X value will be filling shifts to the or the maximum her for the minimum of this, and we're not gonna have any inflection points wouldn't see is bringing a zero. But we can come to the other side to see our inflection point change. So again, when sees it with negative one would get natural log of zero or national one, which would just be zero. So at X equals zero, we have this point of deflection. So to the left of it, it looks concave down. And to the right is conchita and similarly once easy with negative five. Well, wherever our intercept is. That is where the function has this point of inflection. So I would say this describes all the relevant information for this class of functions.