Suppose is a Function of which is twice Differentiable at a Stationary Point .

- 1. If , then has a Relative Minimum at .
- 2. If , then has a Relative Maximum at .

If is a 2-D Function which has a Relative Extremum at a point and has
Continuous Partial Derivatives at this point, then
and
. The second Partial Derivatives test classifies
the point as a Maximum or Minimum. Define the Discriminant as

- 1. If , and , the point is a Relative Minimum.
- 2. If , , and , the point is a Relative Maximum.
- 3. If , the point is a Saddle Point.
- 4. If , higher order tests must be used.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 14, 1972.

© 1996-9

1999-05-26