. . Relative Extrema on an Interval A function f x ( ) is said to have a relative maximum at x , if there is an open interval containing x on which f x is the largest value. Similarly, f x ( ) is said to have a relative minimum at x , if there is an open interval containing x on which f x is the smallest value.
A relative maximum need not be the largest value on the entire domain, while a relative minimum need not be the smallest value on the entire domain. Therefore, there may be more than one relative maximum or relative minimum on the entire domain. A relative extrema of a function is the extreme values (maximum or minimum) of the functions among all the evaluated values of f x I D ( ), ∀∈ ⊂ where I may be open or closed. Usually the local extreme value of a function is attained at a critical point.
Note that, a function may have a critical point at x without having a local extreme value there. For instance, both of the functions y and y have critical points at the origin, but neither function has a local extreme value at the origin. Theorem . (Fermat) If f x ( ) has a relative extrema at x then c is a critical number.
Invariably there will be critical numbers of the function obtained as solutions of the equation ′ or as values of x at which ′ f ( ) does not exist.