Extreme Value Theorem
If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b].
The extreme value theorem is a fundamental concept in calculus that states that a continuous function on a closed and bounded interval attains its maximum and minimum values at least once within that interval.
More formally, if f(x) is a continuous function on the interval [a, b], then there exist points c and d in [a, b] such that:
f(c) ≥ f(x) for all x in [a, b], and
f(d) ≤ f(x) for all x in [a, b].
In other words, f(x) takes its highest and lowest values at some point(s) within the interval [a, b].
The extreme value theorem has important applications in optimization problems, where one needs to find the maximum or minimum value of a function subject to certain constraints. It assures us that a solution exists within a given interval, allowing us to then use other techniques to find the precise value(s) of the maximum or minimum.
It is important to note that the extreme value theorem only holds for closed and bounded intervals, as an unbounded function can technically grow infinitely large without ever attaining a maximum or minimum value.
Moreover, the extreme value theorem is true for both one-dimensional and multi-dimensional functions, meaning that it can be applied to functions with multiple variables as well.
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