local minimum

Examples of local 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

• 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.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • 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 $x$.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Relative Minima and Maxima

  • Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
  • The graph attains a local minimum at $(-1,-2)$ because it is the lowest point in an open interval around $x=-1$. 
  • The local minimum is the y-coordinate $x=-1$which is $-2$.
  • 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.

• Maximum and Minimum Values

  • Similarly, a function has a local minimum point at $x_{\text{min}}$, if $f(x_{\text{min}}) \leq f(x)$ when $\left | x - x_{\text{min}} \right | < \varepsilon$.
  • 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.
  • So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.
  • One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.

• Applications of Minima and Maxima in Functions of Two Variables

  • We have learned how to find the minimum and maximum in multivariable functions.
  • In particular, we learned about the second derivative test, which is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum, using the value of the second derivative at the point.
  • at $\left(\frac{3}{8}, -\frac{3}{4}\right)$ $f(x, y)$ has a local maximum, since $f_{xx} = -\frac{3}{8} < 0$
  • Identify steps necessary to find the minimum and maximum in multivariable functions

• Lagrange Multiplers

  • Sufficient conditions for a minimum or maximum also exist.
  • Every point $\left(\frac{-\pi}{2}, y\right)$$f=-1$ is a global minimum of $f$ with value $-1$.
  • Therefore where the constraint $g=c$ crosses the contour line $f=-1$, is a local minimum of $f$ on the constraint.
  • The trace and the contour $f=-1$ cross at the minimum as we can see in the figure.
  • Since both $g_x \neq 0$ and $g_y \neq 0$, the Lagrange multiplier $\lambda = 0$ at the minimum.

• Other factors affecting supply chain management

  • For example, many countries require that products have a minimum percentage of local content.

• Employment Policy

  • Federal law establishes minimum wages and overtime rights for most workers in the private and public sectors; state and local laws may provide more expansive rights.
  • The FLSA does not preempt state and local governments from providing greater protections under their own laws.
  • Local governments have also adopted a number of "living wage" laws that require those employers that contract with them to pay higher minimum wages and benefits to their employees.
  • This graph of the minimum wage in the United States shows the fluctuation in government guarantees for minimum standards of labor.
  • Illustrate how employment policy is driven by federal, state and local law

• Fair Labor Standards Act

  • 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."
  • The standards apply to employees in the private sector and in Federal, State, and local governments.
  • 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.

• Eligibility

  • Virtually all electoral systems, whether partisan or non-partisan, have some minimum eligibility requirements to run for office.
  • The Senate's minimum age requirement is 30 and nine years an American citizen.
  • For local offices, the requirements are often even less strict — in certain jurisdictions, local officials simply need to be current citizens over the age of 18 who have established local residency.