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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Find the critical points of $f(x, y, z)=x+z$ such that $x^{2}+y^{2}+z^{2}=8$

(-2,0,-2,4),(2,0,2,4)

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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02:48

Use the method of Lagrange…

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01:54

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02:49

And this problem we are asked to find the extreme extreme points are the critical points of F equals X plus C. Subject to the constraint that X squared plus y squared plus B squared minus eight equals zero. And this constraint is a sphere of radius squared to times square to to so we form our augmented function. I call it G. I guess that in the book they call it lambda or L. Anyway, it's an augmented function that with the lagrange multiplier anyway. And so and again I wrote this raw um plus Z. So we have explicit E plus lambert times X squared plus y squared plus b squared minus eight. And so we can take partials partial with respect to X and set the partials to zero is one plus two lambda, X. And that zero partial respect to Y just gives us two lambda. Why that zero respect the Z one plus two lambdas eve that zero. And with respect to land that we just get our constraints and that's zero. Um We can see here that either Y or lambda has to be zero. But you can see also that lambda can't be zero because if it is then we get we get a contradiction 10. So why has to be zero. So and then we can just substitute in and we get to solutions one is X is to y zero, C is too Land is -1 quarter The other is X -2, Y0, Z is -2. And let him just a quarter. And that gives us for this solution here gives us let's see here, that solution there, I write backwards, I might have written that backwards. Um Let's see here too. Yeah, I did, I wrote that backwards. Um this should be four in minus and so I also just kind of confirmed this is that I took, I took this and solved it for Z and then plug it back into here and then you get to branches right, plus or minus the square root of, you know, um eight minus export plus y squared plug that back in here. So I get to two different functions with plus, we'll have a plus or minus in here and you plot that. And this is what we get this kind of surface here, which is kind of like stitch together here. Um I guess it looks kind of like a Mylar balloon, kind of an inflated something. And so we can see here that we have the maximum over here at four and the minimum bet down here at four -4. So again, we can kind of check by embedding the constraint into this and then actually trying to maximize and minimize the surface that results, but you know, at least see where they are. So you can visualize it a bit

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