I think this could act as a proof of min 18:15 claims: * for the degenerate case, we have a line ( x + b / (2a) * y ) = 0 for wich z is allways 0 . It acts like the case of z = x^2 but rotated ( because the term affecting  y^2 is zero in the deg. case ).* for the opposite signs (saddle): if we fix y=0 and let x increase we get possitive values but if we move for the line ( x + b / (2a) * y ) =  0  (the same as for the deg case) we get a decrease in the z values. (because y^2 is positive and the term affecting it is negative for saddle point case).For the conditions of the exercise we start moving from (0,0) and the signs from above are in the case ( 1 / 4a ) is possitive (so change if negative). You can plot some functions on Google: e.g copy paste 6*x**2+5*x*y+y**2 and x**2+4*x*y+4*y**2####[]