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) to analyze its behavior. Since the problem references Example 8, recall the specific function from that example or consider a general approach.
3. Find 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 to find critical points where the function could be constant: solve \(f'(x) = 0\). These points partition the domain into intervals.
5. Determine the intervals between these critical points and check if the function's value remains the same throughout any of these intervals. Since trigonometric functions are continuous and periodic, constant intervals typically occur only at isolated points, so identify if any open intervals exist where the function is constant.

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 Behavior

A function is constant on an interval if its output value does not change for any input within that interval. This means the function’s graph is a horizontal line segment over that interval, and the derivative (if it exists) is zero throughout.

Open Intervals

An open interval (a, b) includes all points between a and b but excludes the endpoints a and b themselves. Identifying the largest open intervals where a function is constant involves finding maximal stretches without including boundary points where the function might change.