Determine the largest open intervals of the domain over which each function is (a) increasing. See Example 9.

Verified step by step guidance

1. Identify the critical points on the graph where the function changes its behavior. In this case, the critical points are (-3, 5) and (0, -4).
2. Observe the behavior of the function between these critical points. The function is increasing when moving from left to right up to the first critical point (-3, 5).
3. Determine the interval over which the function is increasing. From the graph, the function increases from the leftmost point to the peak at (-3, 5).
4. Express the interval in proper notation. The function is increasing on the interval (-∞, -3).
5. Verify that the function does not increase in any other interval. From (-3, 5) to (0, -4), the function is decreasing, and from (0, -4) onwards, the function is constant.

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 at the first. This means that as the input values increase, the output values also increase. Identifying increasing intervals involves analyzing the function's graph or its derivative to determine where the slope is positive.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. In the context of determining increasing intervals, critical points help to delineate where the function changes from increasing to decreasing or vice versa.

Open Intervals

An open interval is a range of values that does not include its endpoints, denoted as (a, b). When discussing the domain of a function, identifying open intervals is crucial for accurately describing where the function is increasing. This means that the endpoints are not part of the interval, which is important when critical points are included in the analysis.