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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y, z)=2 x y+2 x z+y z$ such that $x y z=108$

108

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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Lectures

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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Use Lagrange multipliers t…

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we are asked to find the minimum. Uh This function two X times Y plus two X times E plus Y times Z subject to the constraint that X times Y times Z equals one oh eight or X times Y times Z minus one oh 80. So this is our constraint function. See Alright so again we just have polynomial is but you know they're just their second order or third order and then we wind up with our augmented function equals F plus lambert time. See take the partial with respect to X, Y. And Z. And lambda and set them equal to zero. And we get to Z Plus Y. Times two plus lamb dizzy and that equals zero. And then we get Z plus X times two plus lambda, Z equals zero, Y plus X times two plus lambda Y equals zero. And our constraint equal zero from this derivative. Now again it's probably easiest to solve for lambda in one of these equations back substitute. Get rid of why back substitute, get rid of Z back substitute and we get um get rid of Z. And then we should get an equation for X. And then we can then forward substitute. So it's just a matter of a bunch of algebra and back substitution here. Um They're not letting equations but in the end we actually just find one solution and that is X equals three, Y equals six and Z equals six And it turns out to be -2/3 Plug that into here and we get f because 108 and that should be a minimum um subject to this. Um And obviously we could find some other points that, you know of this constraint, plug them into here and see that they should all be greater than this.

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