TITLE: Why/How does the determinant of the Hessian matrix, combined with the 2nd derivatives, tell us max., min., saddle points? Reasoning behind it?
QUESTION [17 upvotes]: In my classes, we are taught the following.
If the determinant of the Hessian matrix at the critical point $\det(D^2f(c)) > 0$ and $f_{xx}(c) > 0$, the function $f$ at c is concave up.
If the determinant of the Hessian matrix at the critical point $\det(D^2f(c)) > 0$ and $f_{xx}(c) < 0$, the function $f$ at c is concave down.
If the determinant of the Hessian matrix at the critical point $\det(D^2f(c)) < 0$, the function $f$ at c is a saddle point.
However, the reasoning behind this is never explained. We are never taught WHY or HOW.
I would like to know why the determinant of the Hessian matrix, combined with the second derivative at the critical point, contains this information about max., min., and saddle points. I would also like to know how this is derived, as I think this would likely go hand-in-hand with why. 
Please give clear reasoning behind each step - not just 'this is what it is' without any reasoning.
When researching this topic, I recall reading mentions regarding eigenvectors or eigenvalues, but I honestly cannot remember. 
Thank you.

REPLY [2 votes]: You can see it in this way.
Determinant is the product of all eigenvalues of the Hessian matrix (2 eigenvalues, in the case of two variables). Then checking the sign of determinant is sufficient to tell the sign of eigenvalues, which is a more general way to test the min/max points.
FYI: wiki.