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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 and bounded 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: f (c) ≤ f (x) ≤ f (d) ∀ x ∈. {\displaystyle f(c)\leq f(x)\leq f(d)\quad \forall x\in.}

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: m ≤ f (x) ≤ M ∀ x ∈. {\displaystyle m\leq f(x)\leq M\quad \forall x\in.}

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.