Extreme value theorem

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Definition

Suppose f is a continuous function on a closed interval [a,b] (note that f may be defined on a bigger domain, but we are interested in the restriction of f to the closed interval and require it to be continuous). Then, f attains its minimum and maximum value on the interval. In other words, the following statements are true:

  1. Existence of maximum value: There is a real number M such that f(x) \le M for all x \in [a,b] and there exists c_1 \in [a,b] such that f(c_1) = M.
  2. Existence of minimum value: There is a real number m such that f(x) \ge m for all x \in [a,b] and there exists c_2 \in [a,b] such that f(c_2) = m.

Related facts

Applications