Maxima and minima

In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

Definition

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x^∗ if f(x^∗) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x^∗ if f(x^∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

First impression

First impression may refer to:

Minimum wage

A minimum wage is the lowest remuneration that employers may legally pay to workers. Equivalently, it is the price floor below which workers may not sell their labor. Although minimum wage laws are in effect in many jurisdictions, differences of opinion exist about the benefits and drawbacks of a minimum wage. Supporters of the minimum wage say it increases the standard of living of workers, reduces poverty, reduces inequality, boosts morale and forces businesses to be more efficient. In contrast, opponents of the minimum wage say it increases poverty, increases unemployment (particularly among unskilled or inexperienced workers) and is damaging to businesses.

History

"It is a serious national evil that any class of His Majesty's subjects should receive less than a living wage in return for their utmost exertions. It was formerly supposed that the working of the laws of supply and demand would naturally regulate or eliminate that evil [...and...] ultimately produce a fair price. Where... you have a powerful organisation on both sides... there you have a healthy bargaining.... But where you have what we call sweated trades, you have no organisation, no parity of bargaining, the good employer is undercut by the bad, and the bad employer is undercut by the worst.... where those conditions prevail you have not a condition of progress, but a condition of progressive degeneration."