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

Verified step by step guidance

1. Identify the function given in the problem.
2. Understand that a function is constant over an interval if its derivative is zero over that interval.
3. Find the derivative of the function using appropriate differentiation rules.
4. Set the derivative equal to zero and solve for the variable to find critical points.
5. Determine the intervals where the derivative is zero, indicating where the function is constant.

Key Concepts

Function Behavior

Understanding how a function behaves is crucial for determining where it is constant. A function is constant over an interval if it does not change its value within that interval. This means that for any two points in the interval, the function outputs the same value. Analyzing the function's graph or its derivative can help identify these intervals.

Intervals and Domain

In mathematics, an interval is a set of real numbers that contains all numbers between any two numbers in the set. The domain of a function refers to the complete set of possible values of the independent variable. Identifying the largest open intervals where a function is constant involves examining the domain and determining where the function maintains a consistent value without interruption.

Critical Points and Derivatives

Critical points occur where the derivative of a function is zero or undefined, indicating potential changes in the function's behavior. To find intervals where a function is constant, one must analyze its derivative; if the derivative is zero over an interval, the function is constant there. Understanding how to compute and interpret derivatives is essential for identifying these critical points and the corresponding intervals.