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

Verified step by step guidance

1. Identify the function \( f(x) \) whose increasing intervals you need to determine. The problem refers to Example 8, so first write down the explicit form of the function if given, or recall it from the example.
2. Find the first derivative of the function, \( f'(x) \), because the sign of the derivative tells us where the function is increasing or decreasing. Use the rules of differentiation appropriate for the function type (polynomial, trigonometric, exponential, etc.).
3. Set the derivative equal to zero to find critical points: solve \( f'(x) = 0 \). These points divide the domain into intervals where the function's behavior (increasing or decreasing) can change.
4. Determine the sign of \( f'(x) \) on each interval between the critical points by choosing test points in each interval and evaluating \( f'(x) \) at those points. If \( f'(x) > 0 \) on an interval, then \( f(x) \) is increasing there.
5. Collect all intervals where \( f'(x) > 0 \) and express them as the largest open intervals over which the function is increasing. Remember to consider the domain restrictions of the original function when stating these intervals.

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 where the function exists and to analyze its behavior, such as intervals where it increases or decreases.

Increasing Function

A function is increasing on an interval if, for any two points in that interval, the function's value at the larger input is greater than or equal to its value at the smaller input. This concept helps determine where the function rises as the input increases.

Use of Derivatives to Determine Monotonicity

The derivative of a function indicates its rate of change. If the derivative is positive over an interval, the function is increasing there. Analyzing the sign of the derivative helps find the largest intervals where the function is increasing.