Question

suppose that 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 choices.

a. (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.

a. (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 graph of \(y = f^{\prime}(x)\) for increasing intervals of \(f(x)\)

We look for the intervals where \(y=f^{\prime}(x)>0\). From the graph, \(f^{\prime}(x)>0\) when \(x\in(-\infty, - 8)\cup(8,\infty)\)

Step3: Analyze the graph of \(y = f^{\prime}(x)\) for decreasing intervals of \(f(x)\)

We look for the intervals where \(y = f^{\prime}(x)<0\). From the graph, \(f^{\prime}(x)<0\) when \(x\in(-8,8)\)

Answer:

a) The function \(f(x)\) is increasing on the intervals \((-\infty,-8)\cup(8,\infty)\)
b) The function \(f(x)\) is decreasing on the interval \((-8,8)\)