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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y)=x+y$ such that $x y=8$

$-4 \sqrt{2}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

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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And this problem, we want to minimize the sum of X times Y. Subject to the constraint that X times Y. Eight minus eight equals zero. So basically minimized some subject to the product equals eight. For my augmented function, which looks pretty simple, is just G equals X plus Y plus lambda times X, Y minus eight. Take partial with respect to X. Y. And lambda set to micro zero. And we get one plus lambda Y equals zero, one plus lambda X equals zero. And then our constraint equation now um we can, you know, eliminate uh let's see here, we can solve basically this one for lambda, plug that in here. So this one for why plug that into here. And then we get an equation for X and we find that we get to solutions. One solution is when x and Y are both minus square, two times the square to and one solution when X and Y are both plus two times the square there too. So if we want to minimize it We probably be saying we get -4 times the square to and plus four times the square to to for this. So obviously this is are going to be our minimum. So our minimum solution is this point here and this would be I think a maximum

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