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

6912

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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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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Yeah. Now we have F equals X squared. Why is he cubed? And the constraint is a plain X plus Y plus z minus 12 equals zero. Um I don't really know of any interpretation of this but we're Ekstrom izing it and we're trying to find a maximum of this over this plane. So if we look at this plane and we plot this this value on the surface of it, we should find the maximum somewhere. Now. We have our augmented function. He calls x squared plus Y X squared times Y times Z squared cubed plus lambda times this planner. Can this strength constraint here function taking the really expect X. We get lambda plus two X Y Z's cube +40. With respect to Y. We get lambda plus X squared Z cubed. He was zero respect Izzie. We get lambda plus three, X squared Y Z squared Equals zero and respect the land that we just get our constraint equation back. And so you can see we we get a number of solutions and we get a whole actually a whole a different number of solutions here turns out um First of all we get a couple of trivial ones. We're not trivial. Where Yeah, kind of trivial. Where um X zero um Y equals six and Z equals six. Okay, so that's that gives us a zero for this. Get another equipment solution where X zero, Y zero and z is 12 obviously that concern but that makes and this is zero. Then we get a whole funk, a whole line of solutions. Well X zero? Y is 12 minus D N Z is anything. Um, and obviously this is zero for those and then another whole line of solutions. X Texas, anything why equals 12 -100. And again, plugging that into hear anything because this is zero, this is zero. And then we get one other kind of interesting solution that is Excess for Y is two, Z is six. Land that winds up being this -3,456. Now, if we plug that into here, we get f equals 6,912 and that is our maximum of this thing over this plane.

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