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It's been a while since I've looked at multivariable calculus but I'm finding local maxima and minima using a Hessian matrix and I'm stuck at rationalizing the part where if the Hessian is positive and $f_{xx}$ at some point P is negative then f attains a local maximum at P. I think that I see why the Hessian must be positive but why must $f_{xx}$ at P be negative? Any help is greatly appreciated.

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    $\begingroup$ See what it means in the special case in which $f$ is $ax^2+2bxy+cy^2$ and make pictures. (The general case follows from this special one using the 2nd degree Taylor approtimation of the function) $\endgroup$ – Mariano Suárez-Álvarez Oct 22 '16 at 18:21
  • $\begingroup$ So we get the hessian as the discriminant of a general quadratic polynomial. I'm still not completely sure where this is going. $\endgroup$ – user328442 Oct 22 '16 at 18:27
  • $\begingroup$ Make the picture of the graph of my function, and analyze the different cases to see why you need the condition on $f_{xx}$. (In any case, this should be explained in any sensible textbook on calculus with several variables — the very best thing you can do is to browse one0 $\endgroup$ – Mariano Suárez-Álvarez Oct 22 '16 at 18:28

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