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I have got a multivariable function $f$ defined on an open set $V$. Suppose $f$ attains maximum at some point $(x,y)$ inside of $V$. At this point we also have $f_{xx}=f_{yy}=0$. And finally the Laplacian of $f$ on $V$ is greater than or equal to $0$. What can we say? such a point can really exist?

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    We have $V \subseteq \mathbb R^2$, I suppose? Or are $x$ and $y$ multi-dimensional? In either case: Such a point can exist, we can take a constant $f$.2012-04-11
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    For $P=(x_0,y_0)\in\mathbb{R}^2$, $f(x,y)=a[(x-x_0)^2+(y-y_0)^2]+b$ has an extremum (max for $a<0$) at $P_0$ but the Laplacian is $\nabla^2f=4a<0$. So I'd be tempted to say that it's not possible if $f$ is $C^2$ on $V$.2012-04-11
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    What kind of smoothness assumptions are we making about $f$?2012-04-11

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