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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Maximize $f(x, y)=x y$ such that $2 x+6 y=18$

$27 / 4$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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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it was slightly different. Um Take on the first problem. So we have the same function X times Y. But our constraint is slightly different. So we have two X plus six. Y minus 18 equals zero. So we'll call the sea here as I will construct G. By taking F. And adding to Atlanta time. See. Mhm. So are the garage multiplayer lambda now taking partial derivative respect to X. Y. And lambda G gives us to Y. Two lambda plus Y equals zero. This is going to set all these equal to zero to maximize this function. And so then we get partial G. Partial Y is six. Lambda plus X equals zero. And then partial G partial lambda is again just see And that equals zero. So we have um just these three linear equations. Um back substituting, we wind up with X equals nine. Have Y equals we have And land equals -34s. Take this plug it into here and we of course get 27 force and that is a maximum value.

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