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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 $f(x, y, z)=x^{2}+y^{2}+z^{2}$ such that $x-2 y+3 z=6$ and $z-x=2$

$(-1 / 3,-2 / 3,5 / 3,10 / 3)$

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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here. We're trying to find critical points of F equals x squared plus y squared plus z squared. So basically a bunch of nested spheres. So if f week was a constant constant greater than zero, get a sphere. And let's see here, then we have two constraints. And these are two planes again and these two planes were intersect in a line. And then that line is what we're trying to optimize this function over. So optimizing this function subject to the line, that is the intersection of these two planes is the geometrical interpretation. So we have G equals F plus lambda, one, C one plus lambda to see two. So again we have five variables here. So we take the derivative respect to each one of them, X Y, Z, I am the one latitude And set those all equal to zero. You get five equations and five unknowns. But there are linear equations. So we have one solution and you can find them by back substituting or using matrix algebra. Or lots of different ways of finding those. I didn't go through all the ugliness of that. But what you wind up with is X equals minus one third, Y equals minus two thirds, Z equals five thirds. And then these are two lambdas here plugging this back in and we get F equals um 13th. So the Uh the optimal solution would be on a sphere of radius squared 10/3. And we can also basically what I did here to get this plot. As I took these two equations, solve them for X and Y. In terms of Z plug those into here's and then I get a function of Z. So F bar is just a function of Z, and then I plotted it and you can see here that you get this minimum here And it occurs around 10/3 And happens occurs for z equals 5/3 right around here. And then if we found that we could then use their solutions, we got from up here of X and Y in terms of Z to find X and Y. So again, this is consistent with basically solving our constraint equations and then substituting back into our function To eliminate two other variables. So again, it's good to check this kind of to make sure that you're consistent.

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