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For $f(x, y, z)=2 x^{2}+2 y^{2}+3 z^{2}+2 x y+3 x z+5 y z-2 x+2 y+2 z$ find the point(s) at which $f_{x}(x, y, z)=0, f_{y}(x, y, z)=0,$ and $f_{z}(x, y, z)=0$.

$ (1,-1,0)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

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So if we want to find for what point makes the partial derivatives all equal to zero first, we'll just have to find what are the partials? And then what numbers in particular say all those equals zero. So let's just go ahead and do that. First of all, first do tell by Dell X eso this here would be so f sub x is now remember, these wise and these disease we're going to assume are all constants. Eso if we have words just, like squared and then multiplied by a constant Well, that's still a constant. So we take these derivatives, those will be zero. So at least these here Oh, and also that one And then the rest of these We still treat it like it's a constant, Um, but then we can pull it up. So when we do to expert, that would just give us four X over here. So it's plus two x y. So it would be like plus two. Why? Because remember, we can pull that. Why out since we're assuming to Constant and then the derivative of X is just gonna be one, uh, same thing over here. So it just be three z times one and then the derivative of two X is just going to be to be minus two. And now we're going to set this equal to zero, and notice will end up with this system of equations. So I'm gonna add that to over and that is going to be four X plus two y plus three C is equal to two. Okay, So as long as we have this here this relationship between all the values, then our partial derivative with respect to X is going to be equals zero. So what kind of keep this here for now? So let's try to find what are partial with respect to why will yield. So this is going to give us that sub y is equal to. So now the excess and disease we're going to assume are constants. So the ones that air just by themselves, those were just end up being, uh, zero, and then the rest we treat just like their constant so we can factor them out and then take the derivative. Um, but that there should also be zero um, se we would use power also for why, and then we have just plus two, not to I two x and then over here, plus five c and then that would be plus two. And then this is equal to zero. So again, I'll go ahead and subtract the two over. So would be four, actually. Let me go ahead and write this, um, in that same way, we have up there, so would be to X plus four y. Plus, we're actually let me do all this in blue to get pop out. So equals zero. So it would be too X plus or why. Plus five c is equal to negative two. Um And so let's do this one more and then we should have a full system of equations. Then we could just solve that however we want. So now we would do Del by del Z bugs. This would be f sub Z. So now we're going to assume that X and Y are constant. So those are constants. That's all constant. The sex here is a constant This why is a constant at X that wire, Constance So again, if they're just by themselves, then they don't have a Z. Those would all just end up getting? Uh, no, not that one. Um, he's just all get zeroed out, and then we just use power rule or whatever. Well, we need to take the rest of these derivatives, so three z square power rules would be 60 three x z. Remember, three X is a constant. So it's just like we take the derivative Z. So they'll just beat three x times one and then over here, same thing. So five y is a constant. Just take derivatives ease would be five Y, um, time See, four times one. And then over here, that would just be plus two. And then remember, we set this equal to zero, and then I'll write it like we did. The players would be three X plus five wide l six c is equal to now subtract two over negative, too. Okay. And so now let me collect all of these together because now we have a system of equations. So this is going toe hold individually for each of these functions. But since we want all of them to have this at any given moment, we can go ahead and do this all at once. Um, so I'm just gonna set this up as a matrix and then ro reduce it. But, I mean, you could really solve this however you want. Um, I think just doing it via a matrix is a little bit easier. So this would be 356 negative to 4 to 32 245 Negative, too. And then, uh, perfection. Let me plug this in really quickly to see how this world reduces. Looks like we end up with the Matrix. So ones along the diagonals and then one negative. 10 one negative 10 So this implies that if X is, you could have one. Why is equal to negative one and Z is equal to zero. That this here will, um, be the solution to where all of our partial derivatives are zero. So we could just rewrite this as one negative 10 then this here is our solution.

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