maximum

(noun)

The greatest value of a set. 

Related Terms

Examples of maximum in the following topics:

  • Maximum and Minimum Values

    • The value of the function at this point is called maximum of the function.
    • A function has a global (or absolute) maximum point at $x_{\text{MAX}}$ if $f(x_{\text{MAX}}) \geq f(x)$ 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.

  • 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.
    • The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values 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).
    • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
    • If M(a,b)>0M(a,b)>0 and fxx(a,b)<0f_{xx}(a,b)<0, then $(a,b)$ is a local maximum of $f$.
    • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

  • Relative Minima and Maxima

    • A function has a global (or absolute) maximum point at $x$* if $f(x∗) ≥ f(x)$ for all $x$.
    • The local maximum is the y-coordinate at $x=1$ which is $2$.
    • The absolute maximum is the y-coordinate which is $16$.
    • This curve shows a relative minimum at $(-1,-2)$ and relative maximum at $(1,2)$.
    • This graph has examples of all four possibilities: relative (local) maximum and minimum, and global maximum and minimum.

  • 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 maximum 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.
    • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

  • Free Energy and Work

    • The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system.
    • Gibbs energy is the maximum useful work that a system can do on its surroundings when the process occurring within the system is reversible at constant temperature and pressure.
    • The Gibbs free energy is the maximum amount of non-expansion work that can be extracted from a closed system.
    • ΔG is the maximum amount of energy which can be "freed" from the system to perform useful work.

  • Relationship of MC and AVC to MPL and APL

    • There are three points easily identifiable on the TP function; the inflection point (A), the point of tangency with a ray from the point of origin (H) and the maximum of the TP (B).
    • At point A, with LA amount of labour and QA output the inflection point in TP is associated with the maximum of the MP.
    • This maximum of the MP function is associated with the minimum of the MC:
    • At point H, the AP is a maximum at this level of input (LH).
    • Point B represents the level of input (LB) where the output (QB) is a maximum.

  • Adjusting Capacity

    • Capacity adjustment takes into account maximum production levels and the alteration of this level depending on how the firm wants to grow.
    • Adjusting capacity takes into account the maximum level of output that can be produced by a firm, and how that can be changed in order to change the potential forecasts of a firm's performance long term .
    • In the context of capacity planning, "design capacity" is the maximum amount of work that an organization is capable of completing in a given period.
    • "Effective capacity" is the maximum amount of work that an organization is capable of completing in a given period due to constraints such as quality problems, delays, material handling, etc.
    • However, a level of utilization somewhat below the maximum prevails, regardless of economic conditions.

  • Key Points: Range, Symmetry, Maximum Height

    • If you were to draw a straight vertical line from the maximum height of the trajectory, it would mirror itself along this line.
    • The maximum height of a object in a projectile trajectory occurs when the vertical component of velocity, $v_y$, equals zero.
    • Once the projectile reaches its maximum height, it begins to accelerate downward.
    • Like time of flight and maximum height, the range of the projectile is a function of initial speed.
    • Construct a model of projectile motion by including time of flight, maximum height, and range

  • Problems

    • Calculate from the Euler equation and the continuity equation, at what velocity does the flux ($\rho V$) reach its maximum for fluid flowing through a tube of variable cross-sectional area?
    • Estimate the maximum longitudinal fluid velocity in the case of a sound wave in air at STP in the case of a disturbance which sets up pressure fluctuations of order 0.1\%.

  • Financial Applications of Quadratic Functions

    • The method of graphing a function to determine general properties can be used to solve financial problems.Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts.
    • If a financier wanted to find the number of sales required to break even, the maximum possible loss (and the number of sales required for this loss), and the maximum profit (and the number of sales required for this profit), they could simply reference a graph instead of calculating it out algebraically.
    • By inspection, we can find that the maximum loss is $750 (the y-intercept), which is lost at both $0$ and $500$ sales.
    • Maximum profit is $5500 (the vertex), which is achieved at $250$ sales.