Extremum of a Function

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.

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).