One technique for finding local minima or maxima of a function of two variables,f(x,y), uses first-order and second-order partial derivatives.

The technique is as follows:

1. Find all points (a,b) such that f_x(a,b) = 0 and f_y(a,b) = 0. These are called critical points.
2. Compute, for each critical point (a,b), the values
   • A = f_xx(a,b),
   • B = f_xy(a,b),
   • C = f_yy(a,b), and
   • D = AC - B²
3. Consult the following list to find what type of point (a,b) is:
   • If D>0 and A<0, (a,b) is a local maximum.
   • If D>0 and A>0, (a,b) is a local minimum.
   • If D<0, (a,b) is a saddle point.
   • If D=0, this method has failed to identify the nature of (a,b).

Here is an example: Suppose f(x,y) = x³ + y³ - 6xy.

1. f_x(x,y) = 3x² - 6y and f_y(x,y) = 3y² - 6x.

   The functions f_x(x,y) and f_y(x,y) are both zero at (x,y) = (0,0) and (x,y) = (2,2).

2. We note that f_xx(x,y) = 6x, f_xy(x,y) = -6, and f_yy(x,y) = 6y.

   At (0,0), A = 0, B = -6, C = 0, and D = -36.

   At (2,2), A=12, B = -6, C = 12, and D = 108.

3. From the table, we note that (0,0) has D=-36, and so (0,0) is a saddle point.

   We also see that (2,2) has D=108 and A=12, so (2,2) is a local minimum.