Questions: For the function shown in the graph, list the intervals on which the function is increasing, the intervals on which it is decreasing, and the location of all local extrema.

For the function shown in the graph, list the intervals on which the function is increasing, the intervals on which it is decreasing, and the location of all local extrema.

Transcript text: For the function shown in the graph, list the intervals on which the function is increasing, the intervals on which it is decreasing, and the location of all local extrema.

Solution

Solution Steps

Step 1: Find the intervals where the function is increasing.

A function is increasing where the y-value increases as the x-value increases. Looking at the graph, this occurs between x=-4 and x=-2, and also from x=2 onwards. In interval notation, this is written as (-4, -2) and (2, ∞).

Step 2: Find the intervals where the function is decreasing.

A function is decreasing where the y-value decreases as the x-value increases. This occurs from negative infinity up to x=-4, and also between x=-2 and x=2. In interval notation, this is written as (-∞, -4) and (-2, 2).

Step 3: Find the local extrema.

Local extrema are points where the function changes from increasing to decreasing, or vice-versa. At x=-4, the function goes from decreasing to increasing. The y-value at x=-4 is -4, so this is a local minimum at (-4, -4). At x=-2, the function goes from increasing to decreasing. The y-value at x=-2 is 0, so this is a local maximum at (-2, 0). At x=2, the function goes from decreasing to increasing. The y-value at x=2 is -4, so this is a local minimum at (2, -4).