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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y)=2 x+3 y$ such that $x y=24$

-24

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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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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we want to uh Let's see here, is that we want to find the minimum or what we want to find the minimum? Alright. So we want to minimize two x plus three y. Subject to the constraint that X times Y equals 24 or X X times Y minus 24 equals zero augmented function G. Is two, X plus three, Y plus lambert time. See folks see is this partial with respect to X, Y and Z equals zero. We get two plus lambda, Y equals zero, three plus lambda, X equals zero. And then our constraint equation. So we can find um lambda in terms of why here plug that into here and then get X. In terms of why Fuck that into here. And we get an equation for X. And we get to values for X -6 and six. Then the values for Y. When x is minus six, Y is minus four and the nexus six Y. It's four And then lambdas are 1/2 and -1 half. So if we plug this back into here we get -24. And if we plug this back in here we get positive 24. So this is in our minimum solution here. Um Because it's obviously smaller than this, so this is where the minimum occurs at -6 -4.

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