The Extreme Value Theorem: Ensuring the Existence of Maximum and Minimum Values for Continuous Functions on Closed and Bounded Intervals

Extreme Value Theorem

The Extreme Value Theorem is a mathematical theorem that applies to continuous functions on a closed and bounded interval

The Extreme Value Theorem is a mathematical theorem that applies to continuous functions on a closed and bounded interval. It states that if a function f(x) is continuous on the interval [a, b], then it must attain both a maximum value and a minimum value somewhere within that interval.

To understand this theorem, it’s important to know what it means for a function to be continuous. A function is considered continuous if it is defined and does not have any jumps, breaks, or holes in its graph. In simpler terms, you can draw the graph of a continuous function without lifting your pen.

The Extreme Value Theorem guarantees that on a closed interval [a, b], a continuous function will have both a highest point (maximum value) and a lowest point (minimum value). However, it does not specify where these extreme values occur or how many times they may occur.

Let’s illustrate this with an example:
Consider the function f(x) = x^2 on the interval [-1, 1]. This function is continuous over this closed interval, and we can easily see that the maximum value is achieved at x = 1 (f(1) = 1^2 = 1), and the minimum value is achieved at x = -1 (f(-1) = (-1)^2 = 1). Therefore, the Extreme Value Theorem tells us that the maximum and minimum values of f(x) on the interval [-1, 1] are both 1.

It’s worth noting that the Extreme Value Theorem is only valid if the function is continuous and the interval is both closed (has finite endpoints) and bounded (the values within the interval are not infinite or undefined). If any of these conditions is not met, then the theorem does not apply.

The Extreme Value Theorem has applications in various fields, including optimization problems, economics, and engineering, where finding the maximum or minimum value of a function is often crucial. It provides a fundamental mathematical principle that ensures the existence of extreme values for continuous functions on closed and bounded intervals.

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