Second Derivative Test

We conclude this section by stating a Second Derivative Test for maxima and minima of functions of two variables. In practice the test often fails except on small regions D. We therefore have emphasized the tests in the preceding section rather than the Second Derivative Test.

SECOND DERIVATIVE TEST

Suppose z - f(x, y) has continuous first and second partial derivatives on a rectangle D, and (a, b) is a critical point of f in D.

(i) f has a minimum at (a, b) if

at every point of D.

(ii) f has a maximum at (a, b) if

at every point of D.

(iii) f has a saddle point at (a, b) if

For an indication of the proof of the Second Derivative Test, see Extra Problem 55 at the end of this chapter.