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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 $y=2 \sqrt{x}$

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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

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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. As to minimize um X minus four squared plus Y squared subject to the constraint, Y minus two squared rex equals zero. So we formed an augmented function G. Equals F. Plus land at times. See and then we take the partial travelers expect of G. With respect to X. Y. And lambda. It's about the X. We get two X minus lambda over square root of x minus eight. Inspector, why we get lambda plus two Y. Equals zero. And we expect the lander we just get sick was a constraint back and that's always going to be the case now. Um We basically have three equations and three unknowns here and so we can you know, manipulate this a little bit. Say solve this one for why plug it into here so that for lambda, plug it into here and we get an equation for X. And we'll get X equals two and then why he was to times square to and landed his minus four times X. Squared too. Taking these and plugging them into here. We get that. Our minimum value for f. Then is 12

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