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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Maximize $f(x, y, z)=x y z$ such that $2 x+3 y+4 z=36$

72

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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Yeah. Now we're ratcheting things up with three variables in F. So F equals X. Y. X times I can see. And our constraint is basically a plane. And that is three x plus two X plus three Y plus four, Z minus 36 0. So this is a plane. Now. We have, our augmented equation is X times Y. M C plus slammed times are constrained. See constraint function. So basically what we're looking for is the, what are we looking for a maximum of this thing when this plane slices through it? Now, let's see here we have um taking partials with respect to X. Y. Z. And landed. Now. So expect the X. You get to two lambda plus Y times Z. In that equals zero, expect to why we get three lambda plus X times E. And that equals zero. But daisy we get four lambda plus X times Y equals zero and respect the lambda. Again, we just get the constraint equation back now. Um we can just back substitute. And again, these aren't linear. We have four equations and foreign knowns. Mhm. And we can find some kind of trivial solutions where either um were to either or two of x, Y and Z are zero and then the other one satisfies this. Okay, so we get 0090, 12, 0, 18 00. But any of these solutions are going to give this equals zero. So those are those are all ethical zero. And then we have one other solution that isn't kind of trivial. And that is X equals six, Y equals four, Z equals three, and land equals minus six. And we plug this into here And we take that, you know, the product of all those three numbers and we get 72. So that is going to be our maximum value. And I think these are probably minimum, um could be no, probably minimum, but we'd have to check.

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