Understanding The Extreme Value Theorem: Exploring Maximum and Minimum Values of Continuous Functions on Closed Intervals

Theorem 3.1 – The Extreme Value Theorem (3.1)

The Extreme Value Theorem is a fundamental result in calculus that guarantees the existence of both the maximum and minimum values of a continuous function on a closed interval

The Extreme Value Theorem is a fundamental result in calculus that guarantees the existence of both the maximum and minimum values of a continuous function on a closed interval. It can be stated as follows:

Theorem 3.1 – The Extreme Value Theorem:
Let f be a continuous function on the closed interval [a, b]. Then f attains its maximum and minimum values on that interval.

To understand this theorem, let’s break it down:

1. Continuous Function:
A function is said to be continuous if it is defined and has no breaks or jumps in its graph. Intuitively, it means that the graph of the function can be drawn without lifting your pen from the paper. For example, the functions y = x and y = sin(x) are continuous.

2. Closed Interval [a, b]:
An interval is a range of values on the number line. A closed interval includes its endpoints. For example, [0, 1] represents all the values from 0 to 1, including both 0 and 1.

3. Attaining Maximum and Minimum Values:
The Extreme Value Theorem states that for a continuous function f defined on the closed interval [a, b], there exist points c and d in [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]. In other words, the function f reaches its highest and lowest points on the interval.

To illustrate this theorem, consider a function f(x) defined on the interval [0, 2] as f(x) = x^2. The graph of this function is a parabola that opens upwards. By inspection, it is clear that the maximum value of f occurs at x = 2, where f(2) = 4. Similarly, the minimum value of f occurs at x = 0, where f(0) = 0. Thus, the Extreme Value Theorem is satisfied in this case.

The significance of the Extreme Value Theorem is that it guarantees the existence of maximum and minimum values for continuous functions on closed intervals, allowing us to find and analyze these important points in various applications of calculus and optimization problems.

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