Tool to compute extrema of a function. The extremum value of a function is the minimal or maximal value that can take a function.

Extremum of a Function - dCode

Tag(s) : Functions

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An extremum of a function is a point where the function reaches its maximum or minimum.

To find the extreme values of a function (the highest or lowest points on the interval where the function is defined), first calculate the derivative of the function and make a study of the sign.

An extremum of a function is reached when it's derivative is equal to zero and changes of sign.

A minimum of a function $ m $ (lowercase `m`) exists when, for all $ x $, $ f(x) >= m $ is greater than or equal to a minimum $ m $.

__Example:__ Find the extremum of the polynomial $ f(x) = x^2 $ defined over $ \mathbb{R} $: the function has a minimum in $ x=0 $ and $ f(x) >= 0 $ over its domain of definition $ \mathbb{R} $.

A maximum of a function $ M $ (uppercase `M`) exists when for all $ x $, $ f(x) <= M $ is less than or equal to the maximum.

An extremum of a function is necessarily defined on an interval. If the interval is the entire domain of definition of the function then it is a global/absolute extremum, otherwise it is a local/relative extremum.

A function always has 2 global extrema, a maximum and a minimum.

For a constant function, the minimum and the maximum are identical.

The number of local extremums depends on the function, but for a polynomial of degree $ n $, there are, at most, $ n-1 $ local extremums.

__Example:__ The polynomial of degree 2 $ x^2 $ has $ 1 $ local minimum in $ x = 0 $

Extrema is the plural form of extremum (from latin, meaning `Extremity`).

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*Extremum of a Function* on dCode.fr [online website], retrieved on 2024-06-14,

- Absolute Extremum Calculator
- Local/Relative Maximum Calculator
- Local/Relative Minimum Calculator
- What is a function extremum? (Definition)
- How to calculate a extremum?
- What is the difference between a relative/local extremum and an absolute/global extremum?
- How to determine the number of extremum?
- What does extrema mean?

extremum,function,derivative,calculator,maximum,minimum,polynomial

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