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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Find the critical points of $f(x, y, z)=3 x^{2}+4 y^{2}+2 z^{2}$ such that $-x+2 y-3 z=21$ and $2 x-3 y+4 z=15$

(58,-13,-35,13218)

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Use the method of Lagrange…

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um here we want, well given actually to constraint equations and so we have, we want to find extreme points uh critical points of three X squared plus four Y squared plus two Z squared. Which basically sets of kind of nested ellipses. If you set Z equals every quarter constant you get in the lips A constant besides zero. Anyway, now we have one constraint here and another constraint here. These both our planes. Alright, so planes in three dimensional space and these planes they will intersect on a line. And so what we're really doing is we're optimizing finding critical points of this function Over the line that is formed by the intersection of these two planes. So that's what we're doing geometrically. But mathematically we can just form GR augmented function. But now we have to lagrange multipliers. I'm the one of them too because we have two constraints to satisfy. And so we just have, now we have five different variables. Gs a function of X, Y and Z. Lambda. One Atlanta too. So we take the derivatives respect all those variables and settlement crucio. And we get these 123, 45 equations and we have five unknowns. Um They're not it's not too bad because they're all they're all linear. So you can basically set up a matrix if you want or just to kind of back substitution and keep eliminating variables. I didn't go through all the ugly details of that. Um But what you get for a solution is Mexico's 58 Y equals minus 13 Z equals minus 35. And these are our two lambdas Plug that into here. And we get f because 1 13,218. And we could also visualize this by embedding these constraints into here. So I took these two constraints, I solved, solved them simultaneously for X and Y. In terms of Z plug that into here. So now I just have a function of Z. And I can plot that and I get this and we can see here that are minimum Here occurs when z goes amount -35. Um And is it, you know, the F is this, so if I plotted F. Bar which is F with the constraints embedded. And so, you know, if we know what Z is, then we can plug that back into here. And these two we had equations for X and Y from here, in terms of Z. So we can find X and Y. So again, it's just a way of kind of checking confirming our solution here.

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