Proof of Second Partials Test

Question

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac {\partial^{2}f(a,b)}{\partial x\partial y}$$C=\frac {\partial^{2}f(a,b)}{\partial y^{2}}$

1. If $AC-B^{2}>0, A>0:$ $(a,b)$ is a local minimum.
2. If $AC-B^{2}>0, A<0:$ $(a,b)$ is a local maximum.
3. If $AC-B^{2}<0,$ $(a,b)$ is a saddle point.
4. If $AC-B^{2}=0,$ $(a,b)$ is inconclusive.

Best Regards,

Thank You