So we were asked to use a computer algebra system. I use Mathematica here um to find to minimize dysfunction here, F equals X times Y plus Y times Z. Given these two constraints here. And what these constraints are are basically uh Well, they're cylinders, it has two cylinders perpendicular to one, another one with an axis in the Z direction, along with an axis in the Y. Direction. So um let's see here. Well we construct each, so here's our age function. And let's see here we have, you know, it's just a function of X. Y. Z. Lambda one, # two. So we use Mathematica is gradient function and take the gradient with partial with respect to X. Y. Z. We love to one lamb to two. And we get these five equations setting those equal to zero, that we get our two constraint equations back. And then we said these 20 And when we solved that, so I said to make with a zero using this little um kind of notation here, but I'd basically mapped Equals zero over all of these things. So I got an array of equations here and then I can solve them. And we see even if we assume we have real numbers, we can see we get 12345678 solutions. So we have eight solutions and we can see that they're kind of plus or minus. Um they come in kind of plus or minus pairs, Right? So plus plus minus minus, plus minus plus plus, you know? And then minus minus minus plus plus. So and then so some of these are all, there's two at minus Z equals minus 12 as equals one And four z equals zero. So you can kind of see that there's some symmetries here. Right, So all the values are the same or close to the same. Um what did I say minus? No, They get -1. Zeke, was this value and plus or minus minus minus plus plus, um Plus plus minus minus and then our Lagrange multipliers which we really don't care about. Um So we can again here's all of our solutions um and we can we can plug in pulling all these solutions into our function and see what we get and so we get values that either it looks like square to to know that's not square did too. Anyway. Um one plus, quarter to maybe. Um so we get -2.4 Plus 2.4 -2 plus and then we get these .4 ones. Um Again basically too much less than this. Uh So we can see that that our maximum and minimum are here and here, but they actually occur at at two different points. So the first one is this is a maximum here, This X, Y. Z. Here, um that's a minimum. This is a maximum, this is another minimum and another maximum here and then the rest are not. So what's going on here? Well, I tried to make a plot of this, but it gets really ugly. But here we can see kind of what's going on. These are all level sets of X, um X times Y plus Y, times Z. So these are various level sets of those, and I and I plugged them in, I plugged in the solutions. Um and, you know, I plugged in, let's see the level sets I made Yeah, I made level sets of the solutions. And so here's our one cylinder, here's the other cylinder. And then these balls here are where we have solutions um back here. And so what we can see is that obviously these occur on the intersection of the two cylinders and then the, you know, the level set of the function f you know what this value I think is is uhh You know the largest one solution Here, I think that's what that one is. And then we have some other ones like back here. So there's lots of symmetries here and that's what you can see here, that here, the other levels, that's kind of in the interior. Um These are where the other um extremely we got work, but they weren't maximum or minimum. So again, basically they, they all occur on the intersection of these two cylinders and then whenever, you know, the level sets of Hongshan, our largest, and so that we get there and there. And so again, it's kind of hard to see, but that's what's going on here. Um I'm not sure any other physical explanation or geometric explanation for what these are. Level sets are, but what we're looking for is the maximum of these level sets On the intersection of these two cylinders.