Question

the given graph of the derivative f of a function f is shown. assuming the graph continue in the same way as x goes to infinity and negative infinity, answer the following questions. 1. on what intervals is f increasing? answer (in interval notation): 2. on what intervals is f decreasing? answer (in interval notation):

Explanation:

Step1: Recall the relationship

If \(f^{\prime}(x)>0\), the function \(f(x)\) is increasing. If \(f^{\prime}(x)<0\), the function \(f(x)\) is decreasing.

Step2: Identify increasing intervals

From the graph of \(f^{\prime}(x)\), we see that \(f^{\prime}(x)>0\) on the intervals where the graph of \(f^{\prime}(x)\) is above the \(x -\)axis. By observing the graph, \(f^{\prime}(x)>0\) on the intervals \((-4,-2)\cup(2,4)\).

Step3: Identify decreasing intervals

We look for where \(f^{\prime}(x)<0\), i.e., where the graph of \(f^{\prime}(x)\) is below the \(x -\)axis. From the graph, \(f^{\prime}(x)<0\) on the intervals \((-\infty,-4)\cup(-2,2)\cup(4,\infty)\).

Answer:

1. \((-4,-2)\cup(2,4)\)
2. \((-\infty,-4)\cup(-2,2)\cup(4,\infty)\)