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Extreme Value Theorem (Proof) | Maximum and Minimum Values Theorem | Continuity | Advanced Calculus
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EXTREME VALUE THEOREM: In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a maximum and a minimum, each at least once. It is sometimes also known as Maximum and Minimum Values Theorem.
APPLICATION OF THE THEOREM: 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 non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
EXTREME VALUE THEOREM: In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a maximum and a minimum, each at least once. It is sometimes also known as Maximum and Minimum Values Theorem.
APPLICATION OF THE THEOREM: 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 non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
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