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In calculus, the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed interval {\displaystyle } , then f {\displaystyle f} must attain a maximum and a minimum, each at least once. That is, there exist numbers c {\displaystyle c} and d {\displaystyle d} in {\displaystyle } such that:
The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function f {\displaystyle f} on the closed interval {\displaystyle } is bounded on that interval; that is, there exist real numbers m {\displaystyle m} and M {\displaystyle M} such that:
This does not say that M {\displaystyle M} and m {\displaystyle m} are necessarily the maximum and minimum values of f {\displaystyle f} on the interval , {\displaystyle ,} which is what the extreme value theorem stipulates must also be the case.
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 non-empty compact space to a subset of the real numbers attains a maximum and a minimum.