Minimum value of a function

To prove this assertion, it is first useful to consider the mathematical technique of Lagrange multipliers, a method used to find the extreme (maximum or minimum) value of a function subject to constraints. Rather than develop the method in complete generality, we merely introduce it by application to the problem just considered equilibrium in a single-phase, multiple-chemical reaction system. 

Differential calculus can be used to find maximum and minimum values of a function. A relative minimum or maximum value of a variable y which depends on x is found at a point where dy/dx = 0. 

We illustrate the process of finding the maximum and minimum values of a function in an interval in Fig. 4.10. In the interval shown there are three points at which the curve has a horizontal tangent, labeled b, d, and /. The first derivative vanishes at these points. The points at which we might have the maximum value of the function include these three points and the ends of the interval, labeled a and g. At X = / we have a relative minimum, also called a local minimum. At such a point the function has a smaller value than at any other point in the immediate vicinity. At point d we have a relative maximum or a local maximum, at which the function has a larger value than at any other point in the immediate vicinity. The first derivative also vanishes at point b, but this is an inflection point with a horizontal tangent line. 

Sometimes we must find a maximum or a minimum value of a function subject to some condition, which is called a constraint. Such an extremum is called a constrained maximum or a constrained minimum. Generally, a constrained maximum is smaller than the unconstrained maximum of the function, and a constrained minimum is larger than the unconstrained minimum of the function. Consider the following example ... 

One of the most important applications of the differential calculus is the determination of maximum and minimum values of a function. Many of the following examples can be solved by special algebraic or geometric devices. The calculus, however, offers a sure and easy method for the solution of these problems. 

The Determination of Maximum and Minimum Values of a Function by means of Taylor s Series. 

Taylor 8 theorem is sometimes useful in seeking the maximum and the minimum values of a function, say, ... 

This is called Lagrange s criterion for maximum and minimum values of a function of two variables. When this criterion is satisfied f x, y) will either be a maximum or a minimum. To summarize, in order that u = f(x, y) may be a maximum or a minimum, we must have bu... 

The constants must satisfy the following criterion The differences between the observed and the calculated results must be the smallest possible with small positive and negative differences. One of the best ways of fixing the numerical values of the constants in any formula is to use what is known as the method of least squares. This rule proceeds from the assumption that the most probable values of the constants are those for which the sum of the squares of the differences between the observed and the calculated results are the smallest possible. We employ the rule for computing the maximum or minimum values of a function. 

In short, the principle of optimality states that the minimum value of a function is a function of the initial state and the initial time and results in Hamilton-Jacobi-Bellman equations (H-J-B) given below. 

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