Mathematics
Maxima and Minima Problems
Maxima and minima problems in mathematics involve finding the highest (maxima) or lowest (minima) value of a function within a given domain. These problems are often solved using calculus techniques such as finding critical points and using the first or second derivative tests to determine whether a point is a maximum, minimum, or neither.
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5 Key excerpts on "Maxima and Minima Problems"
eBook - ePub- L. Corwin(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
8Maxima and MinimaIn many applications of mathematics, one wishes to maximize (or minimize) some function. In economics, for example, it is usual to try to maximize profit or minimize cost. Finding the extreme values of a function of one variable is one of the applications of elementary calculus. In this chapter, we develop criteria for finding the points at which a function of several variables takes on its maximum or minimum.Just as in the case of functions of one variable, two cases need to be considered. To illustrate, let f : [−1, 1] → R be defined by f(x) = x 2 . The minimum value of f is at x = 0, where f ′(x) = 2x is zero. On the other hand, f has a maximum at x = − 1, but f ′(x ) ≠ 0 at these points. Thus we can use the vanishing of the derivative to find maxima and minima on the “interior” of [ −1,1], but not at the “boundary” points { − 1,1} of [− 1,1]. In this case, of course, there are only two boundary points, while for a function of several variables there are infinitely many. As a result, we need to develop a special procedure for dealing with maxima and minima at boundary points. This procedure, called the method of Lagrange multipliers, is one of the major topics of this chapter. The other is the analysis of maxima and minima at interior points ; as we shall see, this analysis for R n is similar to the case of R 1 .1. Maxima and Minima at Interior Points
Let S be a subset of R n and f : S → R a differentiable function. If S is closed and bounded, we know from Theorems 7.3.3 and 7.4.1 that S has a maximum and a minimum somewhere in S. These theorems, however, give us no information about where the maximum or minimum is. In this section, we discuss a method of locating the maximum and minimum values.We shall solve only a part of the problem here, and we need some definitions to describe what we shall do. We say that a point v ∈ S is an interior point of S if some open ball of positive radius containing v is contained in S ; a point v ∈ S is a boundary point if each open ball containing v meets both S and its complement. (See Exercises 12 to 20 of Section 3.5
eBook - PDFMathematics and Plausible Reasoning, Volume 1
Induction and Analogy in Mathematics
- G. Polya, George Polya(Authors)
- 2020(Publication Date)
- Princeton University Press(Publisher)
• • V l l l • • MAXIMA AND MINIMA Since the fabric of the world is the most perfect and was established by the wisest Creator, nothing happens in this world in which some reason of maximum or minimum would not come to light. —EULER I. Patterns. Problems concerned with greatest and least values, or maximum and minimum problems, are more attractive, perhaps, than other mathematical problems of comparable difficulty, and this may be due to a quite primitive reason. Everyone of us has his personal problems. We may observe that these problems are very often maximum or minimum problems of a sort. We wish to obtain a certain object at the lowest possible price, or the greatest possible effect with a certain effort, or the maximum work done within a given time and, of course, we wish to run the minimum risk. Mathematical problems on maxima and minima appeal to us, I think, because they idealize our everyday problems. We are even inclined to imagine that Nature acts as we would like to act, obtaining the greatest effect with the least effort. The physicists succeeded in giving clear and useful forms to ideas of this sort; they describe certain physical phenomena in terms of minimum principles. The first dynamical principle of this kind (the Principle of Least Action which usually goes under the name of Maupertuis) was essentially developed by Euler; his words, quoted at the beginning of this chapter, describe vividly a certain aspect of the problems on minima and maxima which may have appealed to many scientists in his century. In the next chapter we shall discuss a few problems on minima and maxima arising in elementary physics. The present chapter prepares us for the next. The Differential Calculus provides a general method for solving problems on minima and maxima. We shall not use this method here. It will be more instructive to develop a few patterns of our own instead.
eBook - PDF- Stefan Waner, Steven Costenoble(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 5.2 Applications of Maxima and Minima 401 Graph this function using the technology formula above for 0 # x # h, choosing smaller and smaller values of h, and decide whether f has either a relative maximum or relative minimum at the endpoint x 5 0. Explain your answer. [Note: Very few graphers can draw this curve accurately; the grapher on the Website does a good job (you can increase the number of points to plot for more beautiful results), the grapher that comes with Mac computers is probably among the best, while the TI-83/84 Plus is prob- ably among the worst.] 66. ▼ Refer to Exercise 65, and consider the function f 1 x 2 5 x 2 sin a 1 x b if x ? 0 0 if x 5 0. Technology formula: x^2*sin(1/x) Graph this function using the technology formula above for 2h # x # h, choosing smaller and smaller values of h, and decide (a) whether x 5 0 is a stationary point and (b) whether f has either a relative maximum or a relative minimum at x 5 0. Explain your answers. [HINT: For part (a), use technology to estimate the derivative at x 5 0.] 5.2 Applications of Maxima and Minima In many applications we would like to find the largest or smallest possible value of some quantity—for instance, the greatest possible profit or the lowest cost. We call this the optimal (best) value. In this section we consider several such examples and use calculus to find the optimal value in each. In all applications the first step is to translate a written description into a math- ematical problem. In the problems we look at in this section there are unknowns that we are asked to find, there is an expression involving those unknowns that must be made as large or as small as possible—the objective function—and there may be constraints—equations or inequalities relating the variables.
- Ronald Harshbarger, James J. Reynolds(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Martin Charles Hatch/Shutterstock 627 A P P L I C AT I O N S Advertising, aging workers Water purity, profit, diminishing returns Maximizing revenue, minimizing average cost, maximizing profit Company growth, minimizing cost, postal restriction, inventory-cost models, property development Production costs S E C T I O N S CHAPTER WARM-UP 10.1 Relative Maxima and Minima: Curve Sketching 10.2 Concavity: Points of Inflection Second-derivative test 10.3 Optimization in Business and Economics Maximizing revenue Minimizing average cost Maximizing profit 10.4 Applications of Maxima and Minima 10.5 Rational Functions: More Curve Sketching Asymptotes More curve sketching T he derivative can be used to determine where a func- tion has a “turning point” on its graph so that we can determine where the graph reaches its highest or lowest point within a particular interval. These points are called the relative maxima and relative minima, respectively, and are useful in sketching the graph of the function. The techniques for finding these points are also useful in solving applied problems such as finding the maximum profit, the minimum average cost, and the maximum productivity. The second derivative can be used to find points of inflection of the graph of a function and to find the point of diminishing returns in certain applications. The topics and some representative applications dis- cussed in this chapter include the following. Applications of Derivatives 10 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
eBook - PDFOptimization Techniques
With Applications to Aerospace Systems
- George Leitmann(Author)
- 1962(Publication Date)
- Academic Press(Publisher)
The boundary xz = 0 has two local minima, one at each end of the interval in which the function is defined. It should be noted here that the “suitable neighborhood” for the definition of a local minimuin does not include points outside of the region of definition of the function. The absolute minimum must be found by comparing the values of the local minima D, E, F , G, and H. The basic problem of the theory of ordinary maxima and minima is to determine the location of local extrema and then to compare these so as to determine which is the absolute extremum. The example of Fig. 1 illustrates that a place to look for local extrema is along discontinuities in the first! derivative and also along boundaries (another type of discontinuity).Where the function and its derivatives are continuous the local extema will always occur at stationary points although, as point C illustrates, stationary points are not always local extrema. 1.12 Kecessary Conditions for Maxima or Minima The existence of a solution to an ordinary minimum problem is guar- anteed by the theorem of Weierstrass as long as the function is continuous. This theorem states’: Every function which is continuom in a closed domain possesses a largest and a smallest value either in the interior or on the boundary of the domain. There is no corresponding general existence theorem for the solutions of problems in the calculus of variations, a circumstance which sometimes leads to difficulties. It should be noted that this theorem does not require the derivatives to be continuous so that the theorem applies to problems such as the example of Fig. 1. The location of extrema in the interior of the region may be determined from the following theorem2: A continuous function f(s1, 2 2 , - - *, s,) of n independent variables 21, z2, ..-, xn attains a maximum or a minimum in the interior of a
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