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Find the maximum and minimum values of $f(x,y)=y^2-4x^2$ about the constraint: $x^2+2y^2=4$

When I do the usual Lagrange Multiplier method, the equations come out rather strange: $\nabla f=\lambda \nabla g$ where $g$ is the constraint.

$-8x=\lambda2x$ and $2y=\lambda4y$

which yields different values for $\lambda$. How do I go about solving for the optimization points when this happens?

Any hints would be appreciated.

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  • $\begingroup$ Is there something really basic that I am missing here, seeing the downvotes? $\endgroup$ – user224997 Mar 31 '17 at 4:45
  • $\begingroup$ I am not a downvoter, but your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, consider editing the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ – user409521 Mar 31 '17 at 4:50
  • $\begingroup$ That is what the problem is, there is nothing more to it. $\endgroup$ – user224997 Mar 31 '17 at 4:51
  • $\begingroup$ I don't think that it's a bad question, but if you are not providing details it could look like you are just asking a "homework" question. Most users here prefer that you really want to learn or understand, not just get an answer. That being said, a hint is that $\lambda$ is only different if it is not trivially satisfied. (I.e. this means either $x$ or $y$ is $0$) $\endgroup$ – user345 Mar 31 '17 at 4:55
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    $\begingroup$ Take a look at this question I asked. Very similar problem. math.stackexchange.com/questions/2176281/… $\endgroup$ – user345 Mar 31 '17 at 5:02

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