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least value

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Let f be a real function and D a subset of its domain. If there is a point c in D such that f(c)≤f(x) for all x in D, then f(c) is the least value of f in D. There may be no such point: consider, for example, either the function f defined by f(x)=−1/x, or the function f defined by f(x)=x, with the open interval (0, 1) as D; or the function f defined by f(x)=[x]−x, with the closed interval [0, 1] as D. If the least value does exist, it may be attained at more than one point of D.

That a continuous function on a closed interval has a least value is ensured by the non-elementary theorem that such a function ‘attains its bounds’. An important theorem states that a function, continuous on [a, b] and differentiable in (a, b), attains its least value either at a local minimum (which is a stationary point) or at an end-point of the interval.

Subjects: Mathematics.

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