Question

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

Accepted Answer

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. 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. Find the derivative of the function using standard differentiation rules for trigonometric functions. For example, if the function is $$f(x) = \sin x$$, then $$f'(x) = \cos x. $$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. 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.

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

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Key Concepts

Domain of a Function

Constant Function Behavior

Open Intervals