Question

consider the graph below is the graph of f(x), the derivative of a function f(x). find the open intervals where f(x) is a) increasing, or b) decreasing.
a) list any open interval(s) on which f(x) is increasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
oa. (type your answer in interval notation. use a comma to separate answers as needed.)
b. the function f(x) is never increasing.
b) list any open interval(s) on which f(x) is decreasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
oa. (type your answer in interval notation. use a comma to separate answers as needed.)
b. the function f(x) is never decreasing.

Explanation:

Step1: Recall the relationship between \(f(x)\) and \(f^{\prime}(x)\)

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

Step2: Analyze the sign of \(f^{\prime}(x)\) from the graph

Looking at the graph of \(y = f^{\prime}(x)\):

• For \(f(x)\) to be increasing, we need \(f^{\prime}(x)>0\). The graph of \(y = f^{\prime}(x)\) is above the \(x -\)axis (where \(y>0\)) on the interval \((-\infty,-1)\cup(1,\infty)\).
• For \(f(x)\) to be decreasing, we need \(f^{\prime}(x)<0\). The graph of \(y = f^{\prime}(x)\) is below the \(x -\)axis (where \(y < 0\)) on the interval \((-1,1)\).

Answer:

a) The function \(f(x)\) is increasing on the intervals \((-\infty,-1)\cup(1,\infty)\).
b) The function \(f(x)\) is decreasing on the interval \((-1,1)\).