the given graph of the derivative f of a function f is shown. assuming the graph continue in the same way as…

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):

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):

Answer

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))