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 given in the problem.
2. Find the derivative of the function to determine where it is increasing.
3. Set the derivative greater than zero to find the intervals where the function is increasing.
4. Solve the inequality to find the critical points.
5. Determine the largest open intervals by testing the intervals between the critical points.

Key Concepts

Increasing Functions

A function is considered increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than the value at the first point. Mathematically, if f(x1) < f(x2) for x1 < x2, then the function is increasing. Understanding this concept is crucial for determining where a function rises as you move along the x-axis.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. Identifying the domain is essential when analyzing the behavior of a function, as it can affect where the function is increasing or decreasing. It is important to consider any restrictions, such as division by zero or square roots of negative numbers.

Open Intervals

An open interval is a range of values that does not include its endpoints, denoted as (a, b). When determining the intervals over which a function is increasing, it is important to identify these open intervals, as they indicate where the function maintains its increasing behavior without including the boundary points. This distinction helps in accurately describing the function's behavior across its domain.