Chapter2

CLASSICAL THEORY OF MAXIMA AND MINIMA

• Introduction
• Analytical Methods without Constraints
  • Locating Local Maxima and Minima (Necessary Conditions)
  • Evaluating Local Maxima and Minima (Sufficient Conditions)
  • Sufficient Conditions for One Independent Variables
  • Sufficient Conditions for Two Independent Variables
  • Sign of a Quadratic Form
  • Sufficient Conditions for N Independent Variables
• Analytical Methods Applicable for Constraints
  • Direct Substitution
  • Constrained Variation
  • Lagrange Multipliers
  • Method of Steepest Ascent
  • Economic Interpretation of the Lagrange Multipliers
  • Inequality 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.

    In this example, point A is the global maximum in the region and is located at the top of a sharp ridge. Here the first derivative is discontinuous. A second but smaller maximum is located at point B (a local maximum). At point B the first partial derivatives of y(x1, x2) are zero, and B is called a stationary point. It is not necessary for stationary points to be maxima or minima as illustrated by stationary point C, a saddle point. In this example, the minima do not occur in the interior of the region, but on the boundary at points D and E (local minima). To determine the global minima, it is necessary to compare the value of the function at these points.

    In essence, the problem of determining the maximum profit or minimum cost for a system using the classical theory becomes one of locating all of the local maxima or minima, and then comparing the individual values, to determine the global maximum or minimum. The example has illustrated the places to look which are:

             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.

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