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I'm having trouble with a past exam question regarding the use of Lagrange multipliers for multiple constraints. The question is:

Using the method of Lagrange multipliers for multiple constraints, find the absolute maximum and minimum of $$f(x,y,z) = xy + 2z$$ On the intersection of $$x + y + z =0$$ And $$x^2 + y^2 + z^2 = 24$$ I'm kinda unsure where to start, I haven't really come across many of these types of questions.

Any help would be appreciated.

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You just need to consider $$F=xy+2z+\lambda(x+y+z)+\mu(x^2+y^2+z^2-24)$$ Compute $F'_x,F'_y,F'_z,F'_\lambda,F'_\mu$ and set them equal to $0$.

The same would apply to more constaints. It is just the extension of what you already know and use.

In this particular case where you have one linear constraint, you could eliminate $z$ from it $(z=-x-y)$ and the problem would become $$F=xy-2(x+y)+\lambda(x^2+xy+y^2-12)$$

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  • $\begingroup$ Could you please elaborate a bit more on this? I'm still not really understanding why we compute them like that. $\endgroup$ – Nathan Lowe Mar 28 '17 at 9:57
  • $\begingroup$ @NathanLowe. How do you solve a problem with a single constraint ? Let me know in order I clarify if required. Cheers. $\endgroup$ – Claude Leibovici Mar 28 '17 at 9:58
  • $\begingroup$ Would you use a Lagrange multiplier and attempt to solve for a single variable? $\endgroup$ – Nathan Lowe Mar 28 '17 at 10:06
  • $\begingroup$ @NathanLowe. No. Consider two variables and one constraint. $\endgroup$ – Claude Leibovici Mar 28 '17 at 10:12

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