- published: 23 Sep 2012
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In mathematics, the maximum and minimum (plural: maxima and minima) of a function, known collectively as extrema (singular: extremum), are the largest and smallest value that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).
More generally, the maximum and minimum of a set (as defined in set theory) are the greatest and least element in the set. Unbounded infinite sets such as the set of real numbers have no minimum and maximum.
To locate extreme values is the basic objective of optimization.
A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x∗, if there exists some ε > 0 such that f(x∗) ≥ f(x) when |x − x∗| < ε. The value of the function at this point is called maximum of the function. Similarly, a function has a local minimum point at x∗, if f(x∗) ≤ f(x) when |x − x∗| < ε. The value of the function at this point is called minimum of the function.
Milton Friedman (July 31, 1912 – November 16, 2006) was an American economist, statistician, and author who taught at the University of Chicago for more than three decades. He was a recipient of the Nobel Memorial Prize in Economic Sciences, and is known for his research on consumption analysis, monetary history and theory, and the complexity of stabilization policy. As a leader of the Chicago school of economics, he influenced the research agenda of the economics profession. A survey of economists ranked Friedman as the second most popular economist of the twentieth century behind John Maynard Keynes, and The Economist described him as "the most influential economist of the second half of the 20th century…possibly of all of it."
Friedman's challenges to what he later called "naive Keynesian" (as opposed to New Keynesian) theory began with his 1950s reinterpretation of the consumption function, and he became the main advocate opposing activist Keynesian government policies. In the late 1960s he described his own approach (along with all of mainstream economics) as using "Keynesian language and apparatus" yet rejecting its "initial" conclusions. During the 1960s he promoted an alternative macroeconomic policy known as "monetarism". He theorized there existed a "natural" rate of unemployment, and argued that governments could increase employment above this rate (e.g., by increasing aggregate demand) only at the risk of causing inflation to accelerate. He argued that the Phillips curve was not stable and predicted what would come to be known as stagflation. Friedman argued that, given the existence of the Federal Reserve, a constant small expansion of the money supply was the only wise policy.