>>11924871https://en.wikipedia.org/wiki/Second_partial_derivative_testI honestly love stability questions so I'm happy to help. What you're looking at is a linearization for which we compactly represent in a Hessian matrix. From this matrix, we can find it's eigenvalues and thus make conclusions of the original non-linearized function. The intuition behind why these eigenvalues and derivatives give you the stability is a little more tricky but you can learn that some other time. I personally learned it from a mathematical biology course.
As for the student solution provided, They are missing another step which is to consider the sign of the second derivative of f wrt x. The derivation till step 4 is correct and D < 0. But this only means that our point is a SADDLE. An equivalent condition for saddle is that have DIFFERENT signs. And when they have the same sign, it is either a minimum or a max (depending on D). For our case, let's make sure of this fact:
Thus
https://math.stackexchange.com/questions/406038/second-partial-derivative-test-question this link is also a good read. hopefully these other fags don't discourage you.
tldr; if D<0 it is NOT a minimum, it is a saddle.
