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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Find three numbers whose sum is $S$ if their product is to be as large as possible.

$(\mathrm{S} / 3, \mathrm{S} / 3, \mathrm{S} / 3)$

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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Use Lagrange multipliers t…

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Find by the Lagrange multi…

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well as to basically generalize the previous problems. That's that. We want to maximize the product of three numbers whose sum is S so we go through with all the same steps, right? Hog mental function X equals x times Y times Z slammed equals X plus Y plus z minus S partials with respect to all of the variables X, Y, Z. And lambda. We get basically the same thing as we got before because this is just a constant here. Only in the constraint equation does s show up and so we can we can solve this, this, this and this together. Um, you know, buy back substitution again, we find some kind of trivial solutions to that when and you know, two of their variable too. two of X, Y and Z are zero and the other is S But then F is zero then. And then the other solution we find is that S equals X equals x over three, Y equals x over three, Z equals X over three. And then lambda equals minus X squared over nine. And so our maximum value Given these basically is s cubed over 24, 27. So it's not surprising then that these are all the same. They're always gonna be the same no matter what this is. We're always going to have the same three values. No matter what the some of the of the some what the sum is, we're going to have the same values and that will always be the case. And for these problems

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