4.13 Proof of the Discriminant Test

By Taylor’s Theorem, near to the stationary point $P$ we have an approximation

$$f(a+h,b+k)\approx f(a,b)+{{1}\over{2}}\Bigl(h^{2}{{\partial^{2}f}\over{{\partial x}^{2}}}+2hk{{\partial^{2}f}\over{{\partial x}{\partial y}}}+k^{2}{{\partial^{2}f}\over{{\partial y}^{2}}}\Bigr)_{P}$$

which we can write as

$$f(a+h,b+k)\approx f(a,b)+{{1}\over{2}}\bigl(Ah^{2}+2Bhk+Ck^{2}\bigr)$$

where

$$A=\Bigl({{\partial^{2}f}\over{{\partial x}^{2}}}\Bigr)_{P},\quad B=\Bigl({{\partial^{2}f}\over{{\partial x\partial y}}}\Bigr)_{P},\quad{\hbox{and}}\quad C=\Bigl({{\partial^{2}f}\over{{\partial y}^{2}}}\Bigr)_{P},$$

$$\Delta=AC-B^{2}.$$