Maximum value of a function

This can be done using the Lagrange multiplier procedure for determining the maximum value of a function with constraints. A function V is created as follows ... 

One application of partial derivatives is in the search for minimum and maximum values of a function. An extremum (minimum or maximum) of a function in a region is found either at a boundary of the region or at a point where all of the partial derivatives vanish. A constrained maximum or minimum is found by the method of Lagrange, in which a particular augmented function is maximized or minimized. 

In optimal control, we desire to find the minimum or maximum value of a functional defined over a specified domain. The analytical procedure is to continuously change the associated function from some reference form and examine the corresponding change in the functional. The new form of the function is, in fact, the result of a linear combination of the reference form and some other form of the function in the same domain. This examination can continue only if the new form of the function lies within the specified domain each time the function is changed. Otherwise, the corresponding new functional may not exist or be valid. The validity of the functional is ensured by having the specified domain be a linear or vector space. This space holds within itself all linear combinations of its elements (functions), which are called vectors. A precise definition of linear space is provided in Section 9.19 (p.278). 

This is an even function that has a maximum of 0.399 for x = 0. The function also has two inflection points x — 1 with a corresponding ordinate value of 0.242, which represents 60.6% of the maximum value of the function. The width of the function at the inflection points is approximately 2er (er = 1). In modern chromatography, Wj/2 represents the width of the peak at the half-height (wi/2 = 2.35er) and a2 the variance of the peak. The width of the peak at the base, w, is measured at 13.5% of the height. At this position, if the curve is Gaussian, w = 4a by definition. 

Melendez-Hevia and coauthors then determined the optimum values of t and gc—those that gave the maximum value of a quality function,/, that maximizes GT, CA, and Gp-r, while minimizing... 

In Eqn. (20), Pf corresponds to q, the fraction of broken fibers within a gauge-length of 21 t. Then q, expressed in terms of cr , can be inserted in Eqn. (16), such that the applied stress can be given as function of partial differentiation with respect to Of, the maximum value of a, the tensile strength, ou, can be found ... 

The terms maxima and minima refer to extreme values of a function, that is, the maximum and minimum values that the function attains. Maximum means upper bound or largest possible quantity. The absolute maximum of a function is the largest number contained in the range of the function. That is, if f(a) is greater than or equal to f(x), for all x in the domain of the function, then f(a) is the absolute maximum. For example, the function f(x) = -16x + 32x + 6 has a maximum value of 22 occurring at X = 1. Every value of x produces a value of the function that is less than or equal to 22, hence, 22 is an absolute maximum. In terms of its graph, the absolute... 

A wide variety of problems can be solved by finding maximum or minimum values of functions. For example, suppose it is desired to maximize the area of a rectangle inscribed in a semicircle. The area of the rectangle is given by A = 2xy. The semicircle is given by x + y = r, for y > 0, where r is the radius. To simplify the mathematics, note that A and A are both maximum for the same values of x and y, which occurs when the comer of the rectangle intersects the semicircle, that is, when y = r - x. Thus, we must find a maximum value of the function A = 4x (r -x ) = 4r x - 4x". The required condition... 

When Table V is studied to determine the fuzziness of the image from a point source as a function of the maximum value of a for which even a faint image would be present is about 4.7 mm (d is the minimum distance between the point source of x-rays and the cloth a is the lateral distance in the plane of the cloth from ground zero). When d is 1.0 mm, most of the x-rays reach the cloth, but because of the proximity of... 

The encounter theory is applied to the atom recombination problem in Section VIC. The theory predicts that the recombination rate constant will exhibit a maximum value as a function of bath density. At low densities the rate increases linearly with bath density, then rolls over to exhibit a... 

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. 

The wave number of the fastest-growing disturbance can be obtained by differentiating Eq. (10.4.32) with respect to a for = 0 and setting the result to zero. This calculation was made by Rayleigh, and he found the amplification factor to have a maximum value as a function of a given by... 

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