Critical points and their classification

Each topological feature of p(r), whether it be a maximum, a minimum, or a saddle, has associated with it a point in space called a critical point, where the first derivatives of p(r) vanish. Thus, at such a point denoted by the position vector r, Vp(rj) = 0 where p denotes the operation 

Whether a function is a maximum or a minimum at an extremum is, of course, determined by the sign of its second derivative or curvature at this point. The second derivative of a function/(x) at x (illustrated diagrammati-cally in Fig. 2.2) is the limiting difference between its two first derivatives or tangent lines which bracket that point 

There are just four possible signature values for critical points of rank three. 

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