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
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