In my opinion there is one too many conditions regarding the search for the local maximum of the bivariate Guassian p(x, y). In fact it would not be enough to impose the Hessian determinant of p (x, y) to be > 0, and, using Sylvester's criterion, to impose the top left entry (of the Hessian) to be < 0 too to have the guarantee that both eigenvalues ​​of 'Hessian are negative (ie local maximum condition).
Therefore the condition for which the second derivative of p with respect to y is < 0 is too many.

For more info:
https://math.stackexchange.com/questions/1985889/why-how-does-the-determinant-of-the-hessian-matrix-combined-with-the-2nd-deriva