minimum

(noun)

The smallest value of a set.

Related Terms

Examples of minimum in the following topics:

  • Maximum and Minimum Values

    • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
    • If $M(a,b)>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a local minimum of $f$.
    • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem).
    • Its only critical point is at $(0,0)$, which is a local minimum with $f(0,0) = 0$.
    • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

  • Maximum and Minimum Values

    • The value of the function at this point is called minimum of the function.
    • Similarly, a function has a global (or absolute) minimum point at $x_{\text{MIN}}$ if $f(x_{\text{MIN}}) \leq f(x)$ for all $x$.
    • The global maximum and global minimum points are also known as the arg max and arg min: the argument (input) at which the maximum (respectively, minimum) occurs.
    • Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain.
    • One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.

  • Concavity and the Second Derivative Test

    • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
    • If $f''(x) > 0$ then f(x) has a local minimum at $x$.
    • Now, by the first derivative test, $f(x)$ has a local minimum at $x$.
    • Telling whether a critical point is a maximum or a minimum has to do with the second derivative.
    • If it is concave-up at the point, it is a minimum; if concave-down, it is a maximum.

  • Relative Minima and Maxima

    • The local minimum is the y-coordinate $x=-1$which is $-2$.
    • The absolute minimum is the y-coordinate which is $-10$.
    • This curve shows a relative minimum at $(-1,-2)$ and relative maximum at $(1,2)$.
    • The local minimum is at the $y$-value of−16 and it occurs when $x=2$.
    • This graph has examples of all four possibilities: relative (local) maximum and minimum, and global maximum and minimum.

  • Fair Labor Standards Act

    • The Fair Labor Standards Act of 1938 established a national minimum wage, forbade "oppressive" child labor, and provided for overtime pay in designated occupations.
    • Police officers, fire fighters, paramedics, and other first responders are also entitled to overtime wages and minimum wage pay.
    • It declared the goal of assuring "a minimum standard of living necessary for the health, efficiency, and general well-being of workers."
    • Covered nonexempt workers are entitled to a minimum wage of not less than $7.25 per hour effective July 24, 2009.
    • This graph of the minimum wage in the United States shows the fluctuation in government guarantees for minimum standards of labor.

  • Randomization for two-way tables and chi-square

    • This randomization approach is valid for any sized sample, and it will be more accurate for cases where one or more expected bin counts do not meet the minimum threshold of 5.
    • When the minimum threshold is met, the simulated null distribution will very closely resemble the chi-square distribution.

  • Price Floors

    • By establishing a minimum price, a government seeks to promote the production of the good or service and ensure that the producers have sufficient resources to go about their work.
    • An example of a price floor is the federal minimum wage.
    • History of the federal minimum wage in real and nominal dollars.
    • The federal minimum wage is one example of a price floor.

  • Fair Labor Standards Act

    • The Fair Labor Standards Act (FLSA) established a national minimum wage, "time-and-a-half" for overtime in certain jobs, and etc.
    • The October 26, 1949 Fair Labor Standards Amendment included changes to overtime compensation, defined a "regular rate," redefined the term "produced," raised the minimum wage from 40 cents to 75 cents per hour, and extended child labor coverage.
    • In 1955, the FLSA was amended once again to increase minimum wage, this time to $1 per hour.
    • Subsequent amendments have continued to raise the minimum wage level according to inflation.
    • Several exemptions exist that relieve an employer from having to meet the statutory minimum wage, overtime, and record-keeping requirements.

  • Break-Even Analysis

    • Break-even analysis can determine the minimum amount a company needs to sell in order to cover its costs with no gains or losses.
    • Break-even analysis determines the minimum quantity a company needs to sell in order to cover its minimum costs, including rent, building expenses, utilities, and the operational costs of running day-to-day operations.
    • As long as a business can cover its minimum costs, it is "breaking even" and can remain in business even if it is not turning a profit.
    • Break-even analysis lets companies compare their production or sales with the minimum point (the break-even point) they need to achieve in order to stay in business.

  • Monetary Employee Compensation

    • Many countries dictate the minimum base salary by defining a minimum wage.
    • A map of the United States comparing state minimum wage laws to the federal minimum wage.
    • Kansas is the only state with a minimum wage rate lower than the federal one.