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 first-order and second-order partial derivatives.

The technique is as follows:

1. Find all points (a,b) such that fx(a,b) = 0 and fy(a,b) = 0. These are called critical points.
2. Compute, for each critical point (a,b), the values
• A = fxx(a,b),
• B = fxy(a,b),
• C = fyy(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. fx(x,y) = 3x2 - 6y and fy(x,y) = 3y2 - 6x.

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

2. We note that fxx(x,y) = 6x, fxy(x,y) = -6, and fyy(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.