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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y, z)=x y+2 x z+2 y z$ such that $x y z=32$

48

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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we have see here a function X, Y plus two X plus two Y. Z. And then we have a constraint X Times YMC -32 equals zero. So I'm not sure if a geometrical terms. This is like basically you have a volume of a rectangular solid that it must equal 32. Not sure how you would wind up Good this but we can we can find minimum maximum and we can Ekstrom eyes this thing by Let's see here. Um We can we want to use to find a minimum. Yeah, so you just make G and make that out of F plus lambda time. See take partial with respect to X, Y and Z. And lambda and said to me, go to zero. So we get these four equations. Okay and now um we mean that basically, you know just go back and back substitute and lemonade and try to get everything worked into an equation in 11 equation in one variable and you know that's you can do that many ways. Um Usually it's a good idea to solve for lambda. Um first and then you know, then try to figure out what's what's easier to solve for one variable in terms of another in terms of the others. Anyway, in the end you wind up with a solution that equals to four Xs four Y is for Z is um two and Lambda winds up being -1 And if you plug that in here you get that this function has a value of 48 and that is at least a local minimum. We don't know. It's probably not the global minimum, but it is a local minimum. It may be the global minimum, but we could check basically to see, but it might be it might be um at the global minimum also, but we know that it's it's a local minimum.

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