Suppose  is a function of  that is twice differentiable at a stationary point .

1. If , then  has a local minimum at .

2. If , then  has a local maximum at .

The extremum test gives slightly more general conditions under which a function with  is a maximum or minimum.

If  is a two-dimensional function that has a local extremum at a point  and has continuous partial derivatives at this point, then  and . The second partial derivatives test classifies the point as a local maximum or local minimum.

Define the second derivative test discriminant as

			(1)
			(2)

Then

1. If  and , the point is a local minimum.

2. If  and , the point is a local maximum.

3. If , the point is a saddle point.

4. If , higher order tests must be used.

REFERENCES:

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

Thomas, G. B. Jr. and Finney, R. L. "Maxima, Minima, and Saddle Points." §12.8 in Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, pp. 881-891, 1992.