maximum

(noun)

The greatest value of a set. 

Related Terms

Examples of maximum in the following topics:

  • 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.

  • 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.

  • Scientific Applications of Quadratic Functions

    • The maximum height of a projectile launched directly upwards can also be calculated from a quadratic relationship.
    • The same maximum height of a projectile launched directly upwards can be found using the time at the projectile's peak ($t_h$):

  • Introduction to Inequalities

    • One useful application is in problems that involve maximum or minimum values.
    • Jared has a boat with a maximum weight limit of 2,500 pounds.
    • The inequality states that the total weight of Jared his friends should be less than or equal to the maximum weight of 2,500, which is the boat's limit.

  • The Rule of Signs

    • where $n$ is the total number of roots in a polynomial, $p$ is the maximum number of positive roots, and $q$ is the maximum number of negative roots.
    • Use the rule of signs to find out the maximum number of positive and negative roots a polynomial has

  • Nonlinear Systems of Inequalities

    • Whereas a solution for a linear system of equations will contain an infinite, unbounded area (lines can only pass one another a maximum of once), in many instances, a solution for a nonlinear system of equations will consist of a finite, bounded area.

  • What is a Quadratic Function?

    • where $h$ and $k$ are respectively the coordinates of the vertex, the point at which the function reaches either its maximum (if $a$ is negative) or minimum (if $a$ is positive).

  • Limited Growth

    • The logistic growth model is given by $P(t)=\frac{c}{1+a\cdot e^{-bt}}$ where $P$ represents the present population, $c$ is the carrying capacity (the maximum the population approaches as time approaches infinity), $b$ is the population growth rate, $t$ is time, and $a$ is the difference between carrying capacity and initial population.
    • From the left, it grows rapidly, but that growth is dampened as time passes to where it reaches a maximum.

  • Graphing Equations

    • Parabolas can open up or down, right or left; they also have a maximum or minimum value.

  • Parts of a Parabola

    • If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.