Why does it write all that stuff about fxy2? Isn't it unnecessary? I mean, isn't the following true? If in a point (a,b): fx=fy=0 and fxxfyy>0, then the partial derivatives must be either both negative or positive, and thus point (a,b) is a local minima or maxima. And if fxxfyy<0 => (a,b) is a saddlepoint. Right? So why bring in the fxy2?