Extremum of a Function

To find the extremum 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.

The 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: 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 extremum, otherwise it is a local extremum.

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