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

Verified step by step guidance

1. First, understand that a function is constant on an interval if its value does not change throughout that interval. In terms of trigonometric functions, this means the function's derivative is zero over that interval.
2. Identify the given trigonometric function (e.g., sine, cosine, tangent, or a combination) for which you need to find intervals where it is constant.
3. Compute the derivative of the function using standard differentiation rules for trigonometric functions. For example, if the function is \(f(x) = \sin x\), then \(f'(x) = \cos x\).
4. Set the derivative equal to zero and solve for \(x\) to find critical points where the function could be constant: \(f'(x) = 0\). These points partition the domain into intervals.
5. Analyze each interval between critical points to check if the function remains constant. Since trigonometric functions are continuous and periodic, the only intervals where the function is constant are those where the derivative is zero everywhere, which typically occur at isolated points rather than open intervals. Therefore, conclude that the function is constant only on intervals where it is identically equal to a constant value.

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (usually x-values) for which the function is defined. Understanding the domain is essential to identify intervals where the function behaves in specific ways, such as being constant or increasing.

Constant Function and Constant Intervals

A function is constant on an interval if its output value does not change for any input within that interval. Identifying constant intervals involves finding where the function’s derivative is zero or where the function’s value remains unchanged over an open interval.

Open Intervals

An open interval is a range of values that does not include its endpoints, typically written as (a, b). When determining where a function is constant, it is important to specify open intervals to exclude boundary points where the function’s behavior might change.