Infinite Light Essay

Infinite Light Essay Infinite Light Essay Chandler Meyer Mr.Robel English 2 Honors January 30,2013 Infinite Light What keeps a group of English boys calm and collected on a deserted island with the absence of adults, shelter, or recollection of what happened? In the novel Lord of The Flies by William Golding, the main character Ralph and his wise sidekick Piggy claim a conch shell as their anchor on to civilization. From the beginning Ralph uses the conch to gather and unite all the boys that were on the island. Throughout the story the conch plays a symbolic role representing not only a tie to civilization, but also Ralph’s role as a righteous and justified chief. Towards the end, the conch is thrown into a continuous struggle to preserve civilized manner, or let the savage behavior of Jack’s hunting group dominate the goodness, of which is slipping. From the beginning of the story, the conch plays the role of being the safety net for the boys, as well as a constant reminder that they maintain order and discipline on the island. The importance of the conch is represented right at the start as Ralph and Piggy use it to summon the scattered boys. â€Å"Ralph grasped the idea and hit the shell with air from in his diaphragm. Immediately the thing sounded. A deep, harsh note boomed under the palms, spread through the intricacies of the forest and echoed back from the pink granite of the mountain†(14). Right after the conch is blown boys immediately start to leak in from the lush forest. Once everyone is together and acquainted, there is a meeting held where there is talk among the boys about needing a â€Å"chief†. After comparisons between Jack and Ralph, the more favorable selection points to not only Ralph himself but for the fact that it is he who possesses the conch. As reasons are tossed back and forth there fo r the selection of chief there are some specific ones that were noted about Ralph starting with â€Å"†¦there was his size, and attractive appearance; and most obscurely, yet most powerfully, there was the conch. The being that had blown that†¦was set apart†(19). As the reader can see the conch has displayed already an almost illuminating presence of power from the shell. Right from the moment Ralph claims the position of chief and bearer of the conch, there is a never-ending stab at power and control by Jack as he tries to undermine Ralph, which is present until the very end. Towards the middle of the novel is when Ralph’s authority gets questioned. Along with the organization of the boys little society seeming to slowly evaporate into a regressed state of savagery of which Ralph, Piggy, Simon, and SamnEric seem to avoid. At this point Jack has established a group known as the â€Å"hunters† and is as thirsty for control as ever as he demonstrates when he announces to a group of boys â€Å"We’ll hunt. I’m going to be chief†(123). As events start to fold out and the story continues, the only thing that keeps Ralph holding on to a any link to authority is the conch, but even that seems to be growing less and less of importance in the eyes of Jack’s tribe. In addition to this the reader can also notice that between the two tribes only one is in possession of the conch, which is Ralph's tribe. The symbolization of possessing the conch is that they maintain a civilized structure as well as being able to do things such as maintain a fire and also be able to calmly talk about practical issues that must be addressed at the meetings. In contrast Jack’s tribe, which lacks the conch, is always acting foolish and savage, this could be because they lack the anchor to civilization. Furthermore a great example that shows the crumbling of their society is when Jack leads a plot to steal the fire as he explains to the hunters. â€Å"We’ll raid them and take fire†¦Roger can snatch a branch while I say what I want†(128).This represented really a moral turn for the worst for jack’s group because they had to â€Å"steal† the fire which is an indication of delinquent

Free Essays on Eight Amendment

MY CONSTITUTION PAPER The Constitution is considered the law of the land in the United States of America. It is made up of rules and regulations for the people of the United States to live by. The purpose of the Constitution is to guarantee life, liberty, and the pursuit of happiness for all. The first draft of the Constitution was written in 1787. There have been a few changes and additions made to it, which are called Amendments, since then. I will be focusing on the Eight Amendment which states, "Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishment inflicted" (Constitution of the United States, 1971). In the article I chose from the State News, it told of the ongoing debate about capital punishment. Human rights activists argue with advocates of capital punishment over whether or not the death penalty is cruel and unusual. Is capital punishment breaking the Eighth Amendment? In recent history, the pickax murderer, Karla Faye Tucker, was executed. Human rights activists outside shouted, "Let her live". They believed that every person had a right to live, as stated in the Constitution. They felt that the death penalty was cruel and unusual after Karla had spent 14 years in prison "rehabilitating". Thus, her human rights were violated. In the same manner, supporters of capital punishment shouted, "Kill her, kill her". They believed that the punishment for a violent crime, such as killing two people with a pickax, should be death. In addition, "Advocates hail the death penalty as a factor in falling U. S. crime rates" (The State News). Karla had been given "the right to a speedy and public trial" and the right to appeal, as promised by the U. S. Constitution in the Sixth Amendment. These were her human rights. Up until her death sentence, the Constitution had worked in her favor by keeping her alive. But, the time finally came when she had to accept he... Free Essays on Eight Amendment Free Essays on Eight Amendment MY CONSTITUTION PAPER The Constitution is considered the law of the land in the United States of America. It is made up of rules and regulations for the people of the United States to live by. The purpose of the Constitution is to guarantee life, liberty, and the pursuit of happiness for all. The first draft of the Constitution was written in 1787. There have been a few changes and additions made to it, which are called Amendments, since then. I will be focusing on the Eight Amendment which states, "Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishment inflicted" (Constitution of the United States, 1971). In the article I chose from the State News, it told of the ongoing debate about capital punishment. Human rights activists argue with advocates of capital punishment over whether or not the death penalty is cruel and unusual. Is capital punishment breaking the Eighth Amendment? In recent history, the pickax murderer, Karla Faye Tucker, was executed. Human rights activists outside shouted, "Let her live". They believed that every person had a right to live, as stated in the Constitution. They felt that the death penalty was cruel and unusual after Karla had spent 14 years in prison "rehabilitating". Thus, her human rights were violated. In the same manner, supporters of capital punishment shouted, "Kill her, kill her". They believed that the punishment for a violent crime, such as killing two people with a pickax, should be death. In addition, "Advocates hail the death penalty as a factor in falling U. S. crime rates" (The State News). Karla had been given "the right to a speedy and public trial" and the right to appeal, as promised by the U. S. Constitution in the Sixth Amendment. These were her human rights. Up until her death sentence, the Constitution had worked in her favor by keeping her alive. But, the time finally came when she had to accept he...

Real Estate Competive Corporate Strategy Research Paper

Real Estate Competive Corporate Strategy - Research Paper Example This gives a more comprehensive analysis of how real estate is getting used organization wide. Successful leading organizations put goals and targets, and measure their performance from these tools to ensure continuous improvements. Workplace Business Intelligence gives the organization an ability to compare progress against the set targets by monitoring vital metrics to remove existing organizational inefficiencies. They get used to compare the performance of different locations within the organization (Peoplecube, 2010). Workplace Business Intelligence is not only a real estate management tool, but a service designed to help recognize and remove inefficiencies in little small organizations to global enterprises. Every Workplace Business Intelligence engagement includes professional services not only to help with the installation of the solution, but to help in analyzing and improving the visualizations for the consumers’ different real estate business intelligence data, and develop new visualization required by every organization (Peoplecube, 2010). Porter’s five forces model theory Real estate is a leading is a leading industry in the United States national economy. Porter’s five model analysis tries to reveal the real estate industry and the reasons behind these phenomena. According to the analysis of the relevant state departments, there should be quick development of property management, a real estate industry that is growth oriented from quantity to quality and the growth and upgrading of consumption structure in order to give a high speed real estate industry development (Childress, 2004). Porter developed a structure and on the competitive environment and introduced the five forces model of competition, that is, the five forces model (Porter, 2000). According to Porter, the five forces determine the attractiveness of an industry. These forces include; threat of those who enter the market, threat of the substitute product, buyers ba rgaining power, suppliers bargaining power and the intensity of rivalry. Of The five forces, model determines the degree of competition in an industry; its basic assumption is the profitability of the industry, not only by the attributes of the product but the market structure decision of the industry. The five forces determine the real estate’s competitive strength and profitability. United States Real Estate Five Forces Model The threat of new entrants, buyers bargaining power, threat of substitutes and the intense competition among rivals directly affect the real estate industry and its competitive nature. The different interactions between the five forces determine the levels of industries profits. Threat of new entrants Foreign investors who have access to the to America’s Real Estate industry pose a significant threat. Foreign real estate enterprises and their related constructions and installation will create unprecedented pressure on the domestic enterprises. F or a number, of these industries, the lack of ability to compete will force them to withdraw from the industry, and a large market share of the real industry market will be foreign owned. For most developers in America, this is not a challenge but rather a serious crisis of survival. The new entrants use a lot of resources to eliminate the original product; they provide their product at low prices reducing corporate profits of other real estate firms. Nevertheless, enterprise

Effective Public Speaking Research Paper Example | Topics and Well Written Essays - 1250 words

Effective Public Speaking - Research Paper Example Public speaking has been misunderstood to mean to present an official speech before a broad audience. Having such a notion makes us fail to perfect on our daily conversations that will reflect on a bigger audience and fret when we are requested to carry out a presentation. The expectations we have on people as they present their speech makes us criticize every simple mistake they make. The reason we have such expectations is that of the positions they hold and the influence they have on us. Having the information on such expectations, most known public speakers engage in some form of training and mentorship. They have experts who advise them on how to present a flawless speech. The experts are always available when they are giving the speech and analyze it for the purpose of perfecting their skills (OHair, Friedrich & Dixon, 2007). Most people believe that they will only have to go to such mentorship programs when they are preparing for a major speech, or when their social status has risen and required them to speak frequently to people. However, public speaking is something that we can personally build on and perfect, as we engage in our daily conversations. The fact is, the call to speak to the public may come when least expected and the kind of presentation we make will determine how many more doors will be opened to us. Irrespective of how small or irrelevant the audience may appear to be, a person should always desire to be the best or above others (Grice & Skinner, 2000). On the other hand, it is necessary for a person not to focus on the audience at the expense of what they are presenting it is easier to speak to people comfortably if one understands what they are presenting, rather than when they know whom they are talking to. Surprisingly, people appear comfortable when they are speaking to a particular group and lose themselves when the audience changes.  

Health Econimics and Policy Essay Example | Topics and Well Written Essays - 1250 words

Health Econimics and Policy - Essay Example ining both the advantages and disadvantages of the NHS will lead to a better understanding of the NHS and it may lead to possible ideas for improvement of the NHS. I will begin my discussion of the NHS with its advantages, including its purpose and some of the things that are free. Then consequences of the NHS will be explained, including long waiting times, unfair payments, some parts of health care not being included for free, problems getting enough funding, and perceived similarity to communism. Finally, based on these consequences, some potential solutions will be described. To begin with positive aspects of the NHS, one of the biggest advantages has to do with the reason this organisation was created. The point of the NHS is to give health care to everyone in England, whether they are rich or poor. (Wikipedia, 2008 from NHS website). Instead of health care being a luxury for people with money, the NHS considers it a necessity and a human right. One of the good things about the NHS is that general practitioner visits are free even though some people think they should not be. People need to get referrals from general practitioners to see specialists. â€Å"It has been argued that a nominal charge for an appointment with a GP could be introduced to prevent patients consulting their GP with minor real or imaginary complaints.† (Wikipedia, 2008). If general practitioner visits were not free, though, many people who did not know how much they really needed to go might put their health at risk by skipping the visit to save money. This is a case where it is better to be safe than sorry. Another advantage of the NHS is that it does not require more money from people choosing to do things that make them have more health problems. Some people think that they should not have to help pay for the costs of a smoker who gets lung cancer because that person chose to smoke. (Rodgers, 2003). I am glad that smokers and other people do not have to pay more because

Outsourcing Contract Flaws Essay Example for Free

Outsourcing Contract Flaws Essay In any legal undertaking, contracts play a vital role in ensuring that the interests of both parties are served. Contracts serve as a binding agreement that the signatories should adhere to. In most cases, any violation of the provisions encompassed in a contract holds the erring party liable for breach of contract. When a contract is drafted, the lawyer or whoever is in charge of crafting it has the job of ensuring that both parties understand its provisions and making sure that the contents of the agreement are acceptable to both parties prior to signing the outsourcing agreement.   Ã‚   Consultation with the signatories is a vital requisite in developing a good contract. Outsourcing is one of the newest methods that firms utilize in order to hire personnel. Nowadays, one can easily find firms that specialize in outsourcing services. In this kind of business, an organization employs a service provider to handle various functions such as data entry, programming jobs, and others. Usually located in a foreign country, the outsourcing firm performs jobs that could otherwise be done by the employees of the organization. In a recent survey conducted internationally by the IT Governance Institute (ITGI), it was revealed that clients of outsourcing firms showed a great deal of dissatisfaction with the service provided by these firms. Because of this, the contracting organization made the crucial decision of prematurely severing their ties with the outsourcing firm (Simmonds Gilmour, 2005). In the study, it was further discovered that among the 76 percent of the survey participants providing at least one outsourcing service, only 25 percent have an established method in managing their contracts (Simmonds Gilmour, 2005). The Importance of Contracts in the Outsourcing Business Contracts play an important role in providing outsourcing services. It lays down the duties, roles, goals, and controls needed in anticipating change as well as in handling the development, maintenance, execution, pricing, and regulation of outsourced services(Simmonds Gilmour, 2005). In addition, outsourcing contracts contains provisions that the service provider must follow in order to ensure a general, consistent, and efficient means of handling the outsourcing agreement. Contracts ensure the proper governance of outsourcing (Simmonds Gilmour, 2005). Crafting a contract is by no means an easy job. In order to properly manage outsourcing services, the book entitled Governance of Outsourcing has laid down the following requisites of a good contract:   Outsourcing contracts should indicate what activities are exempted from outsourcing. It should clearly identify the type of service offered. Complicated and permanent outsourcing requirements may require a more integrated approach (Simmonds Gilmour, 2005). Once the outsourcing strategy has been determined, service provider contracts must clearly point out the methods and structures of the outsourcing agreement. It should clearly establish the responsibilities, goals, and roles of the parties involved. This can be formalized through the creation of a governance schedule. Defective approaches should be taken out of the picture. If management is unable to handle such processes, it is unlikely for the outsourcing provider to handle these processes as well (Simmonds Gilmour, 2005). Good outsourcing contracts must have provisions that will clearly evaluate the competence of the people who will be involved in providing outsourcing. Likewise, there should be provisions for checking the credibility and reliability of the chosen service provider (Simmonds Gilmour, 2005). After a year of offering outsourcing, the service provider should work out a re-negotiation of the contract. If the terms of agreement has a global scope and the service is provided to several components of an organization, there should be supporting contracts for every country involved. Contract re-negotiation should be included in the standard operating procedure of the outsourcing firm (Simmonds Gilmour, 2005). Proper channelling of information should be established as this will ensure improved control instead of drawing reactions. Every communication resource should be utilized in order to guarantee that any human resource personnel involved is properly informed (Simmonds Gilmour, 2005). Any transfer of service connected to the agreement must be accompanied by a formal plan and should be acknowledged by the two parties involved in the process. Benchmarking and measurement should likewise be provided in order to properly manage project costing. This is an important aspect in the total conduct of negotiating for a contract (Simmonds Gilmour, 2005). Aside from daily monitoring, the contract should likewise provide for regular benchmarking. Stable governance should be dynamic. Through periodic benchmarking, the service provider will remain valuable and will become the foundation of decisions whether to proceed with contract re-negotiations or abolish the contract (Simmonds Gilmour, 2005). The ITGI survey further revealed that among the organizations included in the study, the main reason why clients outsource is because there are not many people within the organization who have the expertise to perform such service. Forty eight percent of the surveyed organizations support this reason while the remaining 42 percent resort to outsourcing as a cost-reduction measure. However, minimizing costs is not as crucial as improving the value of the organization (Simmonds Gilmour, 2005). There are various benefits that can be derived from outsourcing service. One of the major benefits of this kind of process is better quality of service. Since the organization paid the service provider who specializes in such kind of roles, the firm would have an assurance that the finished product would not be haphazardly done. Another advantage of outsourcing is improved handling of risks. Since the service provider is trained and experienced with the line of work they are paid to do, the risks involved will be much lesser. The outsourcing firm are more equipped to handle possible risks than an employee of the organization who might not have the time to troubleshoot risks. Likewise, there is no need for the organization to select and utilize its own personnel. Instead, the staff of the organization can focus their energy on performing more important and worthwhile activities. The last advantage of outsourcing is scalability. If the organization is satisfied with the performance of the outsourcing firm, it could require its services to perform other needs. For example, a service provider providing data entry functions for a certain organization may be expanded to perform database management duties as well. More and more organizations are realizing the importance of outsourcing as an effective means of strengthening the quality of their business. In order to completely maximize the many advantages of this process, there is a need to ensure the proper governance of these processes.   By drafting stable outsourcing contracts, service providers would be able to address and remedy any possible loopholes that may arise from the outsourcing agreement.  Ã‚   Thus, assuring that their customers are satisfied with the service they provide lies on how effective the outsourcing agreement is crafted by the service provider.   Otherwise, it could lead to a short-term and dissatisfied client.

Maxima and Minima of Functions

Maxima and Minima of Functions This term paper presents concise explanations of the subjects general principles and uses worked examples freely to expand the ideas about solving the problems by suitable methods. Each example shows the method of obtaining the solution and includes additional explanatory techniques. For some topics, where it would have been difficult to understand a solution given on a single problem, the solution has been drawn in step-by-step form. All the figures used have been taken from Google Book search. The term paper covers the necessary definitions on MAXIMA AND MINIMA OF THE FUNCTIONS and some of its important applications. It covers the topic such as types of other method for solving the big problem in a shortcut method known . The aspects of how to develop some of the most commonly seen problems is also covered in this term paper. The motive of this term paper is make the reader familiar with the concepts of application of maxima and minima of the function and  where this is used. Focus has been more on taking the simpler problem so that the concept could be made clearer even to the beginners to engineering mathematics. MAXIMA AND MINIMA The diagram below shows part of a function y = f(x). The point A is a local maximum and the point B is a local minimum. At each of these points the tangent to the curve is parallel to the X axis so the derivative of the function is zero. Both of these points are therefore stationary points of the function. The term local is used since these points are the maximum and minimum in this particular Region. The rate of change of a function is measured by its derivative. When the derivative is positive, the function is increasing, When the derivative is negative, the function is decreasing. Thus the rate of change of the gradient is measured by its derivative, Which is the second derivative of the original function? Functions can have hills and valleys: places where they reach a minimum or maximum value. It may not be the minimum or maximum for the whole function, but locally it is. You can see where they are, but how do we define them? Local Maximum First we need to choose an interval: Then we can say that a local maximum is the point where: The height of the function at a is greater than (or equal to) the height anywhere else in that interval. Or, more briefly: f(a) à ¢Ã¢â‚¬ °Ã‚ ¥ f(x) for all x in the interval In other words, there is no height greater than f(a). Note: f(a) should be inside the interval, not at one end or the other. Local Minimum Likewise, a local minimum is: f(a) à ¢Ã¢â‚¬ °Ã‚ ¤ f(x) for all x in the interval The plural of Maximum is Maxima The plural of Minimum is Minima Maxima and Minima are collectively called Extrema Global (or Absolute) Maximum and Minimum The maximum or minimum over the entire function is called an Absolute or Global maximum or minimum. There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum. Assuming this function continues downwards to left and right: The Global Maximum is about 3.7 The Global Minimum is -Infinity Maxima and Minima of Functions of Two Variables Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points. More on Optimization Problems with Functions of Two Variables in this web site. Theorem Let f be a function with two variables with continuous second order partial derivativesfxx, fyyand fxyat a critical point (a,b). Let D = fxx(a,b) fyy(a,b) fxy2(a,b) If D >0 and fxx(a,b) >0, then f has a relative minimum at (a,b). If D >0 and fxx(a,b) If D If D = 0, then no conclusion can be drawn. We now present several examples with detailed solutions on how to locate relative minima, maxima and saddle points of functions of two variables. When too many critical points are found, the use of a table is very convenient. Example 1:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = 22+ 2xy + 2y2- 6x Solution to Example 1: Find the first partial derivatives fxand fy. fx(x,y) = 4x + 2y 6 fy(x,y) = 2x + 4y The critical points satisfy the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence. 4x + 2y 6 = 0 2x + 4y = 0 The above system of equations has one solution at the point (2,-1). We now need to find the second order partial derivatives fxx(x,y), fyy(x,y) and fxy(x,y). fxx(x,y) = 4 fxx(x,y) = 4 fxy(x,y) = 2 We now need to find D defined above. D = fxx(2,-1) fyy(2,-1) fxy2(2,-1) = ( 4 )( 4 ) 22= 12 Since D is positive and fxx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1). The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6). Example 2:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = 22- 4xy + y4+ 2 Solution to Example 2: Find the first partial derivatives fxand fy. fx(x,y) = 4x 4y fy(x,y) = 4x + 4y3 Determine the critical points by solving the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence. 4x 4y = 0 4x + 4y3= 0 The first equation gives x = y. Substitute x by y in the equation 4x + 4y3= 0 to obtain. 4y + 4y3= 0 Factor and solve for y. 4y(-1 + y2) = 0 y = 0 , y = 1 and y = -1 We now use the equation x = y to find the critical points. (0 , 0) , (1 , 1) and (-1 , -1) We now determine the second order partial derivatives. fxx(x,y) = 4 fyy(x,y) = 12y2 fxy(x,y) = -4 We now use a table to study the signs of D and fxx(a,b) and use the above theorem to decide on whether a given critical point is a saddle point, relative maximum or minimum. critical point (a,b) (0,0) (1,1) (-1,1) fxx(a,b) 4 4 4 fyy(a,b) 0 12 12 fxy(a,b) -4 -4 -4 D -16 32 32 saddle point relative minimum relative minimum A 3-Dimensional graph of function f shows that f has two local minima at (-1,-1,1) and (1,1,1) and one saddle point at (0,0,2). Example 3:Determine the critical points and locate any relative minima, maxima and saddle points of function f defined by f(x , y) = x4- y4+ 4xy . Solution to Example 3: First partial derivatives fxand fyare given by. fx(x,y) = 43+ 4y fy(x,y) = 4y3+ 4x We now solve the equations fy(x,y) = 0 and fx(x,y) = 0 to find the critical points.. 43+ 4y = 0 4y3+ 4x = 0 The first equation gives y = x3. Combined with the second equation, we obtain. 4(x3)3+ 4x = 0 Which may be written as . x(x4- 1)(x4+ 1) = 0 Which has the solutions. x = 0 , -1 and 1. We now use the equation y = x3to find the critical points. (0 , 0) , (1 , 1) and (-1 , -1) We now determine the second order partial derivatives. fxx(x,y) = -122 The First Derivative: Maxima and Minima Consider the function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3   on the interval [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. We cannot find regions of which f is increasing or decreasing, relative maxima or minima, or the absolute maximum or minimum value of f on [à ¢Ã‹â€ Ã¢â‚¬â„¢23] by inspection. Graphing by hand is tedious and imprecise. Even the use of a graphing program will only give us an approximation for the locations and values of maxima and minima. We can use the first derivative of f, however, to find all these things quickly and easily. Increasing or Decreasing? Let f be continuous on an interval I and differentiable on the interior of I. If f(x)0 for all xI, then f is increasing on I. If f(x)0 for all xI, then f is decreasing on I. Example The function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 has first derivative f(x)  =  =  =  123à ¢Ã‹â€ Ã¢â‚¬â„¢122à ¢Ã‹â€ Ã¢â‚¬â„¢24x  12x(x2à ¢Ã‹â€ Ã¢â‚¬â„¢xà ¢Ã‹â€ Ã¢â‚¬â„¢2)  12x(x+1)(xà ¢Ã‹â€ Ã¢â‚¬â„¢2)  Ãƒâ€šÃ‚   Thus, f(x) is increasing on (à ¢Ã‹â€ Ã¢â‚¬â„¢10)(2) and decreasing on (à ¢Ã‹â€ Ã¢â‚¬â„¢Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢1)(02). Relative Maxima and Minima Relative extrema of f occur at critical points of f, values x0 for which either f(x0)=0 or f(x0) is undefined. First Derivative Test Suppose f is continuous at a critical point x0. If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative maximum at x0. If f(x)0 on an open interval extending left from x0 and f(x)0 on an open interval extending right from x0, then f has a relative minimum at x0. If f(x) has the same sign on both an open interval extending left from x0 and an open interval extending right from x0, then f does not have a relative extremum at x0. In summary, relative extrema occur where f(x) changes sign. Example Our function f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 is differentiable everywhere on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], with f(x)=0 for x=à ¢Ã‹â€ Ã¢â‚¬â„¢102. These are the three critical points of f on [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. By the First Derivative Test, f has a relative maximum at x=0 and relative minima at x=à ¢Ã‹â€ Ã¢â‚¬â„¢1 and x=2. Absolute Maxima and Minima If f has an extreme value on an open interval, then the extreme value occurs at a critical point of f. If f has an extreme value on a closed interval, then the extreme value occurs either at a critical point or at an endpoint. According to the Extreme Value Theorem, if a function is continuous on a closed interval, then it achieves both an absolute maximum and an absolute minimum on the interval. Example Since f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 is continuous on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], f must have an absolute maximum and an absolute minimum on [à ¢Ã‹â€ Ã¢â‚¬â„¢23]. We simply need to check the value of f at the critical points x=à ¢Ã‹â€ Ã¢â‚¬â„¢102 and at the endpoints x=à ¢Ã‹â€ Ã¢â‚¬â„¢2 and x=3: f(à ¢Ã‹â€ Ã¢â‚¬â„¢2)  f(à ¢Ã‹â€ Ã¢â‚¬â„¢1)  f(0)  f(2)  f(3)  =  =  =  =  =  35  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢2  3  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢29  30  Ãƒâ€šÃ‚   Thus, on [à ¢Ã‹â€ Ã¢â‚¬â„¢23], f(x) achieves a maximum value of 35 at x=à ¢Ã‹â€ Ã¢â‚¬â„¢2 and a minimum value of -29 at x=2. We have discovered a lot about the shape of f(x)=34à ¢Ã‹â€ Ã¢â‚¬â„¢43à ¢Ã‹â€ Ã¢â‚¬â„¢122+3 without ever graphing it! Now take a look at the graph and verify each of our conclusions. APPLICATION AND CONCLUSION 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) = -162 + 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 maximum of a function is the value of the function that corresponds to the highest point on the graph. Conversely, minimum means lower bound or least possible quantity. The absolute minimum of a function is the smallest number in its range and corresponds to the value of the function at the lowest point of its graph. If f(a) is less t han or equal to f(x), for all x in the domain of the function, then f(a) is an absolute minimum. As an example, f(x) = 322 32x 6 has an absolute minimum of -22, because every value of x produces a value greater than or equal to -22. In some cases, a function will have no absolute maximum or minimum. For instance the function f(x) = 1/x has no absolute maximum value, nor does f(x) = -1/x have an absolute minimum. In still other cases, functions may have relative (or local) maxima and minima. Relative means relative to local or nearby values of the function. The terms relative maxima and relative minima refer to the largest, or least, value that a function takes on over some small portion or interval of its domain. Thus, if f(b) is greater than or equal to f(b  ± h) for small values of h, then f(b) is a local maximum; if f(b) is less than or equal to f(b  ± h), then f(b) is a relative minimum. For example, the function f(x) = x4 -123 582 + 180x + 225 has two relative minima (points A and C), one of which is also the absolute minimum (point C) of the function. It also has a relative maximum (point B), but no absolute maximum. Finding maxima or minima also has important applications in linear algebra and game theory. For example, linear programming consists of maximizing (or minimizing) a particular quantity while requiring that certain constraints be imposed on other quantities. The quantity to be maximized (or minimized), as well as each of the constraints, is represented by an equation or inequality. The resulting system of equations or inequalities, usually linear, often contains hundreds or thousands of variables. The idea is to find the maximum value of a particular variable that represents a solution to the whole system. A practical example might be minimizing the cost of producing an automobile given certain known constraints on the cost of each part, and the time spent by each laborer, all of which may be interdependent. Regardless of the application, though, the key step in any maxima or minima problem is expressing the problem in mathematical terms.