The figure below illustrates the existence of local maxima, local minima, and saddle points for functions of two variables.

As with functions of one variable we want to have efficient criteria to determine the nature of a critical point. For this we look at the second order Taylor approximation of $f(x,y)$ at a critical point $(a,b)$. For a neat function $f(x,y)$ of two variables $x$ and $y$ we have already seen that we can approximate function values near the point $(x,y)=(a,b)$ via $$\begin{aligned}f(x,y) &\approx f(a,b)+f_x(a,b){\vartriangle}x+f_y(a,b){\vartriangle}y\\ &\phantom{=}+ \frac{1}{2}\!\!\left(f_{xx}(a,b){\vartriangle}x^2+2f_{xy}(a,b){\vartriangle}x{\vartriangle}y+f_{yy}(a,b){\vartriangle}y^2\right)\end{aligned}$$ with $${\vartriangle}x=x-a\quad\text{and}\quad {\vartriangle}y=y-b\tiny.$$ When the point $(a,b)$ is a critical point it simplifies to $$f(x,y) \approx f(a,b)+ \frac{1}{2}\!\!\left(f_{xx}(a,b){\vartriangle}x^2+2f_{xy}(a,b){\vartriangle}x{\vartriangle}y+f_{yy}(a,b){\vartriangle}y^2\right)$$

The behaviour of the second order Taylor approximation of $f(x,y)$ at the critical point $(a,b)$ is thus determined by the quadratic function $$h({\vartriangle}x, {\vartriangle}y)=f_{xx}(a,b){\vartriangle}x^2+2f_{xy}(a,b){\vartriangle}x{\vartriangle}y+f_{yy}(a,b){\vartriangle}y^2$$

• If $h({\vartriangle}x,{\vartriangle}y)$ has outside $(0,0)$ only positive values, then $(a,b)$ is a local minimum of the function $f(x,y)$.
• If $h({\vartriangle}x,{\vartriangle}y)$ has outside $(0,0)$ only negative values, then $(a,b)$ is a local maximum of the function $f(x,y)$.
• If $h({\vartriangle}x,{\vartriangle}y)$ has outside $(0,0)$ both positive and negative values, the function $f(x,y)$ has a saddle point at $(a,b)$.

The range of possible values of $h(x,y)$ is closely related to the determinant of the Hesse matrix defined as $$\text{Hesse matrix van }f(x,y) = \left(\begin{array}{cc} f_{xx} & f_{xy} \\ f_{xy} & f_{yy}\end{array}\right)$$ and evaluated in $(a,b)$. The determinant of the Hesse matrix is called the disciminant or the Hessian of ${f(x,y)}$. We denote $$H(x,y)=f_{xx}(x,y)\cdot f_{yy}(x,y)-f_{xy}(x,y)^2$$ and the relationship with $h({\vartriangle}x,{\vartriangle}y)$ is (after completing the square) $$h({\vartriangle}x,{\vartriangle}y) = \frac{1}{f_{xx}(a,b)}\biggl((f_{xx}(a,b){\vartriangle}x+f_{xy}(a,b){\vartriangle}y)^2+{\vartriangle}y^2\cdot H(a,b)\biggr)$$ We can deduce the following criterion: