Washington State University

Department of
Mathematics
Instructor: Dr. Matt Hudelson 
One technique for finding local minima or maxima of a function of two variables,f(x,y),
uses firstorder and secondorder partial derivatives.
The technique is as follows:
 Find all points (a,b) such that f_{x}(a,b) = 0 and
f_{y}(a,b) = 0. These are called critical points.
 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  B^{2}
 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) = x^{3} + y^{3}  6xy.
 f_{x}(x,y) = 3x^{2}  6y and
f_{y}(x,y) = 3y^{2}  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).
 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.
 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.