- Introduction
- Analytical Methods without Constraints
- Analytical Methods Applicable for Constraints
- Necessary and Sufficient Conditions for Constrained Problems
- Closure
- References
- Problems
The classical theory of maxima and minima (analytical methods) is concerned with finding the maxima or minima, i.e., extreme points of a function. We seek to determine the values of the n independent variables x1,x2,...xn of a function where it reaches maxima and minima points. Before starting with the development of the mathematics to locate these extreme points of a function, let us examine the surface of a function of two independent variables, y(x1, x2), that could represent the economic model of a process. This should help visualize the location of the extreme points. An economic model is illustrated in Figure 2-1(a) where the contours of the function are represented by the curved lines. A cross section of the function along line S through the points A and B is shown in Figure 2-1(b), and in Figure 2-1(c) the first derivative of y(x1, x2) along line S through points A and B is given.
1.at stationary points (first derivatives are zero) 2.on the boundaries 3.at discontinuities in the first derivative When the function and its derivatives are continuous, the local extreme points will occur at stationary points in the interior of the region. However, it is not necessary that all stationary points be local extreme points, since saddle points can occur also. |