University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 11. Extrema of Multivariable Functions

11.1  Critical Points

a.  Function has a local maximum (or minimum) on , if there exists a neighborhood of such that for all of that neighborhood there holds

b.   which is a local maximum or minimum point is called a point of local extremum of .

c.  Necessary Condition for the Existence of Extremum: if function has an extremum on , and, in addition, it has first-order partial derivatives at that point, then all these derivatives equal zero on .

d.   is a critical point of if all partial derivatives of equal zero at .

e.  All critical points and points at which at least one partial derivative does not exist are suspected extrema points.

11.2  Types of Critical Points

If is a critical point of function belonging to class (it has continuous partial derivatives up to second order), and the quadratic form (see next paragraph).

of variables is

1.  Positive, then function has local minimum on .

2.  Negative, then function has local maximum on .

3.  Mixed, then function has no extremum on .

11.3  Analysis of Quadratic Forms

The matrix of quadratic form (see XI.12) is

and

are k first minors of matrix A.

Sylvester Theorem: The form is positive if, and only if, all first minors of matrix are positive. Quadratic form is negative if, and only if, the sign of its first minors change alternately, the first sign being negative.

11.4  Extremum of a Two-Variable Function

Function has a local extremum on in which , , and . If , then it is the local minimum, and if , then it is the local maximum. If , function has no extremum on . In such a case, is a saddle point.