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I would like to find the extrema of the function $f(x,y)=x^2+4xy+4y^2$ subject to $x^2+2y^2=4$ using Lagrange Multipliers.

Is it possible to get for the Lagrange multipliers the value zero?

I don't think so because the gradient vectors for $f$ and $g(x,y)=x^2+2y^2$ have to be proportional.

So this problem has two solutions. But their image by the function $f$ is the same. So, how can I know if they correspond to a maximum or to a minimum?

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