And this problem we are asked um what we're given the problem where we want to find the minimum distance between this function here or G one equals zero and G two equals zero. So I plotted those here and G1 equals zero is just a line here. And this is kind of like a rotated problem but rotated um in the plane. So that again, Z here is corresponding to um corresponding to why for this curve and w corresponding to X. No other way around that makes it here. What did I say that? Right? Um Yes, he is correspondent to Why W correspondent to X. Yeah, that makes sense. And so you know, we won't use Lagrange multipliers to form this new function age and then the rest of it I did in Mathematica because that's just what I'm familiar with. Obviously pretty much any computer algebra system should be able to do this. So, you know the notation maybe a little unfamiliar, but hopefully you can, I'll explain what's going on here. So if you have to find F as a function of X, Y, W and Z. So what we're doing is f is actually the distance between two points, any two points on these two on two paths. And so what we're doing is trying to find a minimum distance. And so I define G one MG too, that I defined H. So H is a function of X. Y. W. Z. Lambda one lambda too. And then I took the gradient. So in Mathematica there's this gradient function and you give it the function you want to take the gradient of and then the parameters um that you want to differentiate with respect to. And so that gives us a list or an array of 1, 2, 3, 4, 5 6 equations that will depend on X Y W. Ramsey number one and number two. So we have six equations and six unknowns that we need to solve. So what I did here is this is just a little little thing why I basically set every one of these components of this factor um to zero. So we have now we have six equations, all this grouped into an array. And then I asked, I asked Mathematica assault mathematical to solve them and just look for real solutions. And so it was able to spit out some nice clean answers. And so we can see that on the one curve, The point is -18 78. So that's this point here. And then on the other curve there parabolic curve we have ah let's see here. This is X, w correspondence to X is one quarter And Z is 1/2. So that's this point here. And so the distance is 9/32 units. So that's um when we plug in, plug in all these this solution into um into this function, the distance function. And so what we see is that this is the line connecting the two points. And interestingly what you can see is that um this line is perpendicular to both of these curves and I think, I can't remember, I think that's always going to be the case when when you have the minimum between any two points because yeah, I can't remember exactly why that is, but I'm pretty sure that's the case that that any time you find the distance between the minimum of the distance between two curves, that the line connecting those points will be perpendicular to both curves. I'm pretty sure that's the case anyway. Um So again, you can see here we just minimized and obviously this looks looks right. Um This is kind of like if you think about how, how the distance is changing as you move away from this point, it certainly seems like the distance between any two points. You know, it's going to be growing, you know, So obviously this is not good or you know, this is not so anyway, um it looks about right and this is your solution.