Questions: Use the graph to determine the open intervals over which f(x) is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.

Use the graph to determine the open intervals over which f(x) is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.

Transcript text: Use the graph to determine the open intervals over which $f(x)$ is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.

Solution

Solution Steps

Step 1: Identify Intervals of Increase

The function $ f(x) $ is increasing where the graph is moving upwards as $ x $ increases. From the graph, $ f(x) $ is increasing on the intervals:

$ (-\infty, -2) $
$ (0, 2) $

Step 2: Identify Intervals of Decrease

The function $ f(x) $ is decreasing where the graph is moving downwards as $ x $ increases. From the graph, $ f(x) $ is decreasing on the intervals:

$ (-2, 0) $
$ (2, \infty) $

Step 3: Determine Local Minimum and Maximum Values

Local minimum and maximum values occur at the points where the graph changes direction from increasing to decreasing or vice versa. From the graph:

Local minimum at $ x = -2 $ and $ x = 2 $
Local maximum at $ x = 0 $

Final Answer

Intervals of increase: $ (-\infty, -2) $ and $ (0, 2) $
Intervals of decrease: $ (-2, 0) $ and $ (2, \infty) $
Local minimum values at $ x = -2 $ and $ x = 2 $
Local maximum value at $ x = 0 $

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