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$f(x,y)=12x^2+12y^2+(x+y)^3$

Find all local maxima, minima and saddle points.

I found $2$ critical points $(0,0)$ and $(-2,-2)$ but the second derivative test came out weird for the set of points $(-2,-2)$. I got $f_{xx}=0$ and the discriminant bigger than $0$. That doesn't imply anything (not maxima, not minima, not saddle point and test isn't even inconclusive). What do I do at this point?

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I would say the test is conclusive: the discriminant is negative (-24^2) so it is a saddle.

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  • $\begingroup$ Forgot that $f_{xy}$ was squared. Thank you! $\endgroup$ – Omrane Dec 14 '15 at 22:29
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I would say it's a saddle point as well. Here's a quick plot of the function at the point in questionenter image description here

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