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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Show that $L_{\lambda}=0$ always yields the constraint equation $c=0$.

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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asked us to show that the augmented function they call it el And with the derivative specter lambda. So they want us to show that in their notation, L lambda equals C equals zero. Um yield that the constraint equal zero, Z equals zero. And so I've been basically using that fact for and basically we should have realized that at some point that that's always going to be the case after all these example problems. Um but generally speaking what we can say is that we have, you know, our function is going to be some function of some set of variables X one to X. N however many there are. And then our constraint is going to be some function of our constraint equate or constraint function. See there's going to be a function of all those variables to possible but none of these are gonna be this is not a functional plan that this is not a functional founder. So the only land a that appears is right here. So when you take the parcel delivery respect to lambda, this is a constant. That this is a constant and we just get see back and so and basically setting that equal to zero, just gives us the gift of constraint equation That C equals zero. So if we write our constraint function, you know like this, something like that a lot of times we'll have some other things like you know because some constant. Okay, say whatever. Um but we can always take this and move everything to one side and then get a function C equals zero. And so we can put that into here. And then basically what happens is when we do the partial derivatives and do our optimization Alright, I find are critical points and we take the dream of the respective lambert. We basically embed the constraint again, so that you can realize that every problem you solved one of the, so one of the 34 or five equations that you had to find the variables. One of them was this this equation here at the constraint equation. And so that is always going to be the case that the partial with respect to lambda of our augmented function, there's always gonna be this thing here, see which was as our constraint function. And so we're then we're gonna, when we set that equal to zero, we're gonna get our constraint equation back again. So again, that's always going to be the case and I basically used that through all all these problems, but hopefully, you know, generally speaking, lambda is never going to appear and effort or see. It's kind of this extra variable we had in, it only appears linearly

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