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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Maximize $f(x, y, z)=x y z$ such that $36 x^{2}+9 y^{2}+4 z^{2}=1$

$\frac{\sqrt{3}}{324}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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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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we want to maximize the product of X times Y times X. Y. And Z. Subject to the constraint um 36 X squared plus nine Y squared plus four Z squared minus one equals zero. This isn't a lips and lips away And three D. So over the surface we want to find the maximum of the product of those of the three coordinates. So that's kind of the geometric interpretation of this. So we set G equal to X times Y. M. C. Plus lambda times are constraint function which constrains us to be on this ellipse. Oid now we have, let's see here we have you take the revenue respect to X, Y and Z. And lambda. And said to me, go to zero. So we wind up with 72 lambda times X plus Y times Z equals zero, 18. Lander times, Y plus X times equals zero, X times Y plus eight. Lambda times equals zero. And then our constraint equation here when we take them to respect the lambda. Now we wind up with a whole bunch of solutions here. I think there are, let me see here, I need to I don't have my, I think that there's 1, 2, 3, 456. And then I think there's let's see, are we taking out, I think all combinations of these. Um So there's a ridiculous number of solutions to this. Let me see how many there actually are. Ah um Where are we here, I look through my notes, it's two constraints. I Where is this 1? Oh yeah right here we have 123456789, 10, 11, 12, 13, 14. Yeah, so all different combinations of these things, so um lots of solutions here. Now, what we can see is that obviously any of these solutions give f equals zero. And now these solutions depends on, you know, the signs of these things. We always get one over 108 times the square to three, but in some cases we get plus, so if all of them are plus, if two of them are minus and one is plus, um we get the positive sign, if all of them are minus would get the negative sign. Two of them are plus and one of them is minus. We get the negative sign. So we get all kinds of combinations of here, so to get the maximum, obviously we need the plus sign, so we need the sign Sign of each of these terms to be a part 3 1. You know, if we take SGN so if this is plus, this is plus and this is plus, then we have that, this is minus, this is minus and this is plus, then we have a plus sign if this is minus and this is minus and this is plus, we have the plus sign. So we have a whole set of solutions. Um and again, that's just because of kind of the symmetries in this ellipse. Oid and then this function also, so we have lots and lots of solutions and there are lots and lots of maximum um whenever the signs of these three functions of their product is is one.

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