Determine the largest open intervals of the domain over which each function is b) decreasing. See Example 9.

Verified step by step guidance

1. Step 1: Identify the critical points on the graph where the function changes from increasing to decreasing or vice versa. From the graph, these points are at \(x = -3\) and \(x = 0\), with coordinates \((-3, 5)\) and \((0, -4)\) respectively.
2. Step 2: Understand that a function is decreasing on intervals where the graph moves downward as \(x\) increases. This means the slope of the function is negative in those intervals.
3. Step 3: Observe the graph between the critical points. From \(x = -3\) to \(x = 0\), the graph is moving downward, indicating the function is decreasing on the interval \((-3, 0)\).
4. Step 4: Check the behavior of the function outside these points. To the left of \(x = -3\), the graph is increasing, and to the right of \(x = 0\), the graph is constant (horizontal line), so the function is not decreasing there.
5. Step 5: Conclude that the largest open interval where the function is decreasing is \((-3, 0)\).

Key Concepts

Decreasing Function

A function is decreasing on an interval if, as the input values increase, the output values decrease. Graphically, this means the curve moves downward as you move from left to right within that interval.

Open Intervals

An open interval is a range of values that does not include its endpoints. When identifying intervals where a function is decreasing, we focus on open intervals to exclude points where the function might change behavior, such as local maxima or minima.

Critical Points and Local Extrema

Critical points occur where the function's slope is zero or undefined, often corresponding to local maxima or minima. These points help determine where the function changes from increasing to decreasing or vice versa, which is essential for identifying decreasing intervals.