Maximum and minimum values of a function

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. 

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. 

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... 

Find maximum and minimum values of a function of one variable ... 

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. 

If/(a) is a one-to-one function, but not necessarily monotonic, then the maximum and minimum values of the function need to be determined over the given region, that is, the following 3 steps need to be performed ... 

In the case of planar jr-bonded groups, the maximum and minimum values of the steric parameter are available from the van der Waals radii (Figure 2). Both the steric effect and the delocahzed electrical effect are a function of the dihedral angle when this type of group is bonded to an sp hybridized carbon atom of a skeletal group . ... 

Repeating such computations for other values of p within the range of instability, we can see how the waveform, amplitude, and period vary with the reactant concentration. Figure 2.6(a) plots the maximum and minimum values of the concentration of A as a function of p. This representation emphasizes how the system oscillates about the unstable stationary state... 

The effective bulk dielectric constant is determined by measuring the distance between a maximum and minimum value of amplitude. The bulk loss factor is determined by measuring the amplitude of the signal under the sample with the loop, as a function of plunger distance from the beginning of the sample. 

This amount of uncertainty disappears when the residence time of the observed tracer element is known. Next, the RTD function for the maximum and minimum values of H(t) are discussed mathematically. Since the variable t takes a positive value and the average of t is fixed at unity, according to Section 1.6, H(t) assumes maximum and minimum values as... 

Step 6. Steps 3- are repeated until a predefined convergence criterion is achieved. For the convergence criterion one can use the difference between the maximum and minimum values of the fitness function. Calculations are terminated when this difference falls below a certain threshold (e.g., 0.02). 

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... 

There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and engineering. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business it is usually possible to express profit as a function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In... 

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 ... 

Fig. 23.12 (a) Incremental decrease in the contact radius as a function of the difference in the maximum and minimum values of the energy release rate for each cycle of the fatigue experiment. Data are shown for tests with three different 80°C contact times. 

Figure 5.9 Maximum and minimum values of the corrosion rate in carbonated concrete as a function of environmental humidity [10]...

Since the sine function has maximum and minimum values of 1 and -1, respectively, jc in (4.24) oscillates between A and —A. The sine function repeats itself every 2tt radi-... 

There are basically two reasons for this distortion. Standard differencing schemes introduce numerical diffusion to the initial distance function. This problem can be reduced by using higher-order numerical methods. The second reason is that the flow field is very rarely of that character so that the level set function cj) would be kept as a distance function. For example, the maximum and minimum values of the level set function will remain the same throughout the computations. For two merging interfaces (e.g., two bubbles), this will cause a steep gradient and an impenetrable sheet between the two merging interfaces (see Fig. 9). It is therefore necessary to reinitiate the distance function after each time step. 

Since the sine function has maximum and minimum values of 1 and —1, respectively, X in (4.22) oscillates between A and —A. The sine function repeats itself every 2tt radians, and the time needed for one complete oscillation (called the period) is the time it takes for the argument of the sine function to increase by 2tt. At time t + /v, the argument of the sine function is 2Trv t + /v) + b = 27rvt + 2tt + b, which is 2tt greater than the argument at time t, so the period is l/v. The reciprocal of the period is the number of vibrations per unit time (the vibrational frequency), and so the frequency is v. 

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