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

$7 / 4$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 4

The Method of Lagrange Multipliers

Partial Derivatives

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Harvey Mudd College

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University of Nottingham

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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we were asked to, let's see here minimize this function here um X minus two squared plus y squared are subject to the constraint that y minus x, Y minus squared of x equals zero. So here's our constraint. So we form our augmented function here, x minus two squared plus y squared plus land at times our constraint time. See here we take partial of the revolution back to X, Y. And lambda and set them all equal to zero. And so with respect to X we get two x minus four minus lambda over squared facts. Um For a while we get lambda plus two, Y equals zero. And we expect Atlanta we just get see And that equals zero. Which we you know that basically just gives us our constraint equation back now. Uh we can you know, solve this from for why? Well actually these are all all um Oh no they're not all separable. But um anyway we can solve this phone for why put it into here, find X. And put that in here. And so anyway, after some back substitution we have three equations and three unknowns. After some algebra we wind up with X equals three halves, Y equals square root of three halves. And then the equals minus six. Find a square with a six plugging these things back into here says that F equals seven Force. And so that is a minimum. And we could check that by plugging other points into here and seeing just other random points and seeing that they're all bigger than this

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