Extremal tests for scalar functions of several real variables at degenerate critical points

Abstract

It is, of course, well known to students of calculus that the extremal nature of a function f at a critical point is decided if the so-called discriminant (or Hessian) given by the expression A =f2-fxxfyy evaluated at the critical point is nonzero. It is also known through simple examples that, in the so-called degenerate case when A = 0 at the point, f may have either an extremum or a saddle point. Consequently, the extremal nature of fin this case is indeterminate from a knowledge of the second derivatives at the point in question alone and higher order partial derivatives at the point must be considered. (It is interesting that during the last century a certain confusion existed concerning the degenerate case, even apparently in the minds of some renowned mathematicians. For a short account of this history of the degenerate case see [1].) Systematic, yet straightforward and simple methods by which to take into account the higher order derivatives seem, however, difficult to come by. In fact, the only method known to the author which offers an essentially complete account of this case is due to Freedman [2]. (His techniques are concerned with the solution of the equationf (x, y)= 0 for x = x (y) but implicitly yield information about extrema as well. He also considers cases other than the degenerate case. Also in a recent paper I-3] Butler and Freedman consider the case when the lowest order terms off are cubic or higher; as stated below, we do not consider this case here.) The purpose of this note is to present a complete method for determining the extremal nature of f on the basis of its derivatives at the point in question under the two assumptions that (i)f possesses the necessary number of partial derivatives and (ii) the lowest order terms in its Taylor expansion with remainder at the point are quadratic. Under these conditions we will show how the extremal nature off may be decided in the degenerate case through a sequence of tests each involving a discriminant and each having a degenerate case, whose occurence, however, can be followed by the next test of the sequence. Each test has the same format as the standard discriminant test using A, which itself may be considered as simply the first test of the sequence. Although they accomplish more or less the same ends, the details