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Use the method of Lagrange multipliers to optimize $f$ as indicated, subject to the given constraint(s).Minimize $f(x, y)=(x-4)^{2}+y^{2}$ such that $y=2 \sqrt{x}$

108

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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Yeah. So as to minimize two xy subject to the constraint that Y -27 plus X squared equals zero. These are constraint equation or the sea. Set up our augmented function two X. Y. So F. Plus lambda time. See here and then we're gonna take partial driven those respect to X. Of disrespect to X. Y. And Z. And lambda and set them equal to zero. So we get to land at times X plus two Y equals zero. Then we get lambda plus two, X equals zero. And then our constraint equation here. And so in this case we see that you know we can back substitute and get an equation. Let's see here itself here for X. Let's see here. That would probably be yeah. So this one for X. I can't remember how exactly I did this. Maybe I saw this one for why? And this one for lambda and then plug them back into here and got an equation for X Turns out that X equals plus or -3. So both of those solutions work in either case lambda is our Why is 18 and Lambda is plus or -6. So if you took these two solutions and plug them back into here we get f equals plus or -118. So if we wanted to minimize that then obviously -1, 18 is the minimum. And that's the case when we get when we have um minus three and 18. So our solution then is really minus 3 18 for the minimum

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