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If $\bigtriangledown^2f = 0 $ in some region in the space, then $f$ cannot have maximum or minimum on that region.

My approach was to assume $f$ has a maximum and then use the second derivative test to obtain a contradiction. Is this a right approach? Is there an easy way to tackle this problem?

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It's hard to know if your approach is right, since you didn't give any of the details. – Nate Eldredge Nov 1 '12 at 3:43
haha i don't wanna be 'that guy' but do you want $f$ nonconstant, and the region connected :p? – uncookedfalcon Nov 1 '12 at 4:46

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