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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y)=\sqrt{x^{2}+y^{2}}$ such that $3 x+4 y=12$

2.4

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Oregon 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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well as to minimize the square root of X squared plus y squared, subject of the constraint that three X plus four Y minus 12. And that should equal zero. Yeah. All right. So augmented functions squared of X squared plus Y squared plus lambda times are constraint? See partially respect to X. You get three lambda plus X divided by a squared of X squared plus y squared. Set that equal to zero? Partial respect. Why we get four lambda plus? Why over squared of X squared plus Y squared? And we set that equal to zero partial with respect to lambda. We could see and then we set that equal to zero. So um again, it's just a matter of uh back substituting and trying to eliminate, we can kind of eliminate um Landau from these two equations fairly easily. And then, you know, eliminate another variable. And in fact in this case we only get one solution. We get that x equals 36/25 and y equals um 48/25. And Landry wind up being one minus 1/5. And so we plug this back into here, we get that this f is um 12, All over five

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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject…