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Extremum of a Function

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

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Extremum of a Function -

Tag(s) : Functions

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# Extremum of a Function

## Local/Relative Minimum Calculator

### How to calculate a extremum?

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 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.

### What is the difference between a relative/local extremum and an absolute/global extremum?

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.

### How to determinate the number of 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 $x = 0$

### What does extrema mean?

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

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