Those points consist of interior domain points where f ' ( x)= 0, interior domain points where f ' does not exist, and the domain's endpoints, which are not covered by the theorem.Ī critical point is an interior point in the domain of a function at which f ' ( x) = 0 or f ' does not exist. Because of Theorem 2, only a few points need to be considered when finding a function's extreme values. It does not say that every point where the first derivative equals zero must be a local extremum. Theorem 2 says that if a function has a first derivative at an interior point where there is a local extremum, then the derivative must equal zero at that point. Theorem 2 If a function has a local maximum value or a local minimum value at an interior point c of its domain and if f ' exists at c, then f ' ( c) = 0. Theorem 2 below, which is also called Fermat's Theorem, identifies candidates for local extreme-value points. So, absolute extrema can be found by investigating all local extrema.Ĭandidates for Local Extreme-Value Points It is clear from the definitions that for domains consisting of one or more intervals, any absolute extreme point must also be a local extreme point. The definition can be extended to include endpoints of intervals.Ī function f has a local maximum or local minimum at an endpoint c of its domain if the appropriate inequality holds for all x in some half-open interval contained in the domain and having c as its one endpoint. The definition of local extrema given above restricts the input value to an interior point of the domain.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |