Theorem 3.5: Test for Increasing and Decreasing Functions – A Comprehensive Guide to Determine Function Behavior on Specified Intervals

Theorem 3.5 – Test for Increasing and Decreasing Functions (3.3)

Theorem 3

Theorem 3.5 (Test for Increasing and Decreasing Functions) is a theorem that provides a method to determine whether a given function is increasing or decreasing on a specified interval. This theorem is also known as the first derivative test.

Let’s assume that we have a function f(x) defined on an interval (a, b) and f'(x) represents its derivative. The theorem states:

1. If f'(x) > 0 for all x in (a, b), then f(x) is increasing on (a, b).
2. If f'(x) < 0 for all x in (a, b), then f(x) is decreasing on (a, b). 3. If f'(x) = 0 for all x in (a, b), then f(x) is constant on (a, b). In simple terms, the theorem states that if the derivative of a function is positive on an interval, then the function is increasing on that interval. Similarly, if the derivative is negative, then the function is decreasing. If the derivative is zero, then the function is constant. To apply this theorem: 1. Differentiate the given function f(x) to find its derivative f'(x). 2. Determine the interval on which you want to test the increasing or decreasing behavior of the function. 3. Evaluate the derivative f'(x) on the given interval, and analyze its sign. 4. If the derivative is positive for all points in the interval, the function is increasing on that interval. If the derivative is negative, the function is decreasing. If the derivative is zero for all points, the function is constant. It is important to note that this test applies to open intervals (a, b) and may not provide conclusive information about the behavior of the function at the endpoints of the interval. To analyze the behavior at the endpoints, additional methods, such as the second derivative test, may be required. This theorem is a powerful tool in analyzing the behavior of functions and can be used to determine where a function is increasing, decreasing, or constant on a given interval. It is often used in calculus and real analysis to study the properties of functions.

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