least and greatest value of function
Theorem.
If the real function f is
-
1.
continuous
on the closed interval
[a,b] and
-
2.
differentiable
on the open interval (a,b),
then the function has on the interval [a,b] a least value and a greatest value. These are always got in the end of the interval or in the zero of the derivative
.
Remark 1. If the preconditions of the theorem are fulfilled by a function f, then one needs only to determine the values of f in the end points a and b of the interval and in the zeros of the derivative f′ inside the interval; then the least and the greatest value are found among those values.
Remark 2. Note that the theorem does not require anything of the derivative f′ in the points a and b; one needs not even the right-sided derivative in a or the left-sided derivative in b. Thus e.g. the function f:x↦√1-x2, fulfilling the conditions of the theorem on the interval [-1, 1] but not having such one-sided derivatives, gains its least value in the end-point x=-1 and its greatest value in the zero x=0 of the derivative.
Remark 3. The least value of a function is also called the absolute minimum and the greatest value the absolute maximum of the function.
Title | least and greatest value of function |
Canonical name | LeastAndGreatestValueOfFunction |
Date of creation | 2013-03-22 15:38:57 |
Last modified on | 2013-03-22 15:38:57 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 26B12 |
Synonym | global extrema of real function |
Related topic | Extremum![]() |
Related topic | LeastAndGreatestNumber |
Related topic | FermatsTheoremStationaryPoints |
Related topic | MinimalAndMaximalNumber |
Defines | absolute minimum |
Defines | absolute maximum |