Mathematics
Maxima and Minima
Maxima and minima refer to the highest and lowest values of a function, respectively. In calculus, finding these points is important for optimization problems, where the goal is to find the maximum or minimum value of a function within a given range.
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6 Key excerpts on "Maxima and Minima"
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 - ePubIntroduction to Differential Calculus
Systematic Studies with Engineering Applications for Beginners
- Ulrich L. Rohde, G. C. Jain, Ajay K. Poddar, A. K. Ghosh(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
greater than the maximum of the function at x = x 1 .Figure 19b.1At this stage, we introduce the following terms and concepts that will be frequently used in this chapter.- Absolute Maximum (Minimum) of a Function : In Figure 19b.1 , note that at x = b , the value of the function is greater than any maximum of the function on the interval [a , b ]. Thus, the greatest value of the function occurs at x = b , and similarly, the smallest value occurs at x = x 2 . We say that the absolute maximum of f is f (b ) and the absolute minimum is f (x 2 ).
- The Points of Extreme Values of a Function : The points like x 1 , x 2 , x 3 , x 4 , and b at which the extreme values of the function f occur , are called the points of extremum (or extreme) values of the function. Note that, a function defined on an interval can reach maximum and minimum values only for the points that lie within the given interval .1
Our interest lies in finding the points of extreme values of a continuous function by using the concept of the derivative . Once such points are known, it is easy to compute the extreme values of the function and then select the absolute extreme values, which have practical applications , as will be clear from some solved examples.In the case of some functions, it is not difficult to find the points of extrema without using calculus, but it will be seen that in general it is not possible to find the extreme values without applying differential calculus.2The knowledge of such points is very useful in sketching the graph of a given function. Besides, these extreme values have many practical applications in widely varying areas such as engineering and various sciences, and so on.
eBook - ePubTwo and Three Dimensional Calculus
with Applications in Science and Engineering
- Phil Dyke(Author)
- 2018(Publication Date)
- Wiley(Publisher)
Chapter 3 Maxima and Minima 3.1 Introduction In the last chapter, Taylor's theorem in two variables was proved. In this chapter, we look at two-dimensional surfaces that have extreme values. In fact we shall find points where the local gradients are zero then determine whether these points are extreme. The general name for these are turning points as it is quite possible for gradients to be zero yet the points be neither maximum nor minimum. However the name extreme has become generic and synonymous with turning point. Taylor's theorem in two dimensions is used to derive criteria for classifying whether an extreme point is a minimum, maximum or a saddle point. Consider a function of two variables written where in this chapter and are the standard three-dimensional Cartesian co-ordinates. Such a function therefore represents a surface in space. For each pair of co-ordinates, defines a value, and as and vary this traces out a surface. It is assumed that the function is well defined so that only one value of corresponds to a given pair of co-ordinates. Therefore, for example, a whole sphere could not be represented, but the top hemisphere could. In this chapter, this is not a problem; but it is a bit of a nuisance when multiple integrals are considered later and its resolution can wait until then. Definition 3.1 A function is said to have a relative minimum at the point if there is a disc centred at the point such that for all points that lie inside this disc. Figure 3.1 The function pictured with some transparency has a relative minimum at the point Definition 3.2 A function is said to have a relative maximum at the point if there is a disc centred at the point such that for all points that lie inside this disc. At a minimum or a maximum the tangent plane is horizontal. Visualise this plane as a small flat square
- 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
eBook - PDF- Deborah Hughes-Hallett, Andrew M. Gleason, Daniel E. Flath, Patti Frazer Lock, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale, Jeff Tecosky-Feldman, Joseph Thrash, Karen R. Rhea, Thomas W. Tucker(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
We use the adjective “local” because we are describing only what happens near . Local Maxima and Minima are sometimes called local extrema. How Do We Detect a Local Maximum or Minimum? In the preceding example, the points = −2 and = 8, where () = 0, played a key role in leading us to local Maxima and Minima. We give a name to such points: For any function , a point in the domain of where () = 0 or () is undefined is called a critical point of the function. In addition, the point (, ()) on the graph of is also called a critical point. A critical value of is the value, (), at a critical point, . 192 Chapter 4 USING THE DERIVATIVE Notice that “critical point of ” can refer either to points in the domain of or to points on the graph of . You will know which meaning is intended from the context. A function may have any number of critical points or none at all. (See Figures 4.3–4.5.) Critical point Figure 4.3: A quadratic: One critical point Figure 4.4: () = 3 + + 1: No critical points Critical points ❄ ❄ ❄ 1 −1 Figure 4.5: () = sin : Many critical points Geometrically, at a critical point where () = 0, the line tangent to the graph of at is horizontal. At a critical point where () is undefined, there is no horizontal tangent to the graph— there is either a vertical tangent or no tangent at all. (For example, = 0 is a critical point for the absolute value function () = , but is not differentiable at = 0.) However, most of the functions we will see are differentiable everywhere, and therefore most of our critical points will be of the () = 0 variety. The critical points divide the domain of into intervals on which the sign of the derivative remains the same, either positive or negative. Therefore, if is defined on the interval between two successive critical points, its graph cannot change direction on that interval; it is either going up or going down.
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