|Washington State University
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:
- Find all points (a,b) such that fx(a,b) = 0 and
fy(a,b) = 0. These are called critical points.
- 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
- 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
Here is an example: Suppose f(x,y) = x3 + y3 - 6xy.
- fx(x,y) = 3x2 - 6y and
fy(x,y) = 3y2 - 6x.
fx(x,y) and fy(x,y) are both
zero at (x,y) = (0,0) and (x,y) = (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 =
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
We also see that (2,2) has D=108 and A=12, so (2,2) is
a local minimum.