Determine the largest open intervals of the domain over which each function is (b) decreasing. See Example 8.

Verified step by step guidance

1. Step 1: Identify the function. If the function is not provided, assume a general trigonometric function such as f(x) = sin(x), cos(x), or tan(x).
2. Step 2: Find the derivative of the function, f'(x), to determine where the function is increasing or decreasing.
3. Step 3: Set the derivative f'(x) < 0 to find the intervals where the function is decreasing.
4. Step 4: Solve the inequality from Step 3 to find the critical points and determine the intervals.
5. Step 5: Verify the intervals by testing points within each interval to ensure the function is indeed decreasing in those intervals.

Key Concepts

Function Behavior

Understanding how a function behaves is crucial in determining its intervals of increase and decrease. A function is said to be decreasing on an interval if, for any two points within that interval, the function's value at the first point is greater than its value at the second point. This behavior can often be analyzed using the first derivative test.

First Derivative Test

The first derivative test is a method used to determine where a function is increasing or decreasing. By calculating the derivative of the function, we can identify critical points where the derivative is zero or undefined. Analyzing the sign of the derivative in the intervals around these points allows us to conclude whether the function is increasing or decreasing in those intervals.

Open Intervals

Open intervals are ranges of values that do not include their endpoints, denoted as (a, b). When determining where a function is decreasing, it is important to express these intervals correctly, as including endpoints can change the nature of the function's behavior at those points. Identifying the largest open intervals ensures that we capture the full extent of the function's decreasing behavior without including points where it may not be decreasing.