The Extreme Value Theorem: Guaranteeing Maximum And Minimum Values For Calculus And Real Analysis With Practical Applications In Economics, Physics, And Engineering

Extreme Value Theorem

f is continuous on [a,b]f has an absolute minimum and an absolute maximum on [a,b]

The Extreme Value Theorem is a fundamental theorem in calculus and real analysis which states that if a function f is continuous over a closed and bounded interval [a, b], then f attains both a maximum and a minimum value on [a, b].

In other words, the Extreme Value Theorem guarantees that there exist points c and d, in the interval [a, b] such that f(c) is the maximum value of f on [a, b], and f(d) is the minimum value of f on [a, b].

The proof of the Extreme Value Theorem is based on the concept of the completeness of the real numbers, which essentially says that a non-empty set of real numbers that is bounded above (or below) has a least upper bound (or greatest lower bound) that belongs to the set.

The Extreme Value Theorem has important applications in various fields such as optimization, economics, physics, and engineering. It is used to find the maximum and minimum values of a function over a given interval, which can be useful in solving real-world problems such as maximizing profits or minimizing costs in business applications, or determining the maximum and minimum values of physical quantities such as velocity or acceleration in physics and engineering.

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