In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval, then f must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in such that:
A related theorem is the boundedness theorem which states that a continuous function f in the closed interval is bounded on that interval. That is, there exist real numbers m and M such that:
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.
The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a compact space to a subset of the real numbers attains its maximum and minimum.
Read more about Extreme Value Theorem: History, Functions To Which Theorem Does Not Apply, Extension To Semi-continuous Functions, Topological Formulation, Proving The Theorems
Famous quotes containing the words extreme and/or theorem:
“Airplanes are invariably scheduled to depart at such times as 7:54, 9:21 or 11:37. This extreme specificity has the effect on the novice of instilling in him the twin beliefs that he will be arriving at 10:08, 1:43 or 4:22, and that he should get to the airport on time. These beliefs are not only erroneous but actually unhealthy.”
—Fran Lebowitz (b. 1950)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)