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 choice.

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: Determine when \( f(x) \) is increasing

A function \( f(x) \) is increasing when \( f^{\prime}(x)>0 \). Looking at the graph of \( y = f^{\prime}(x) \), we see that \( y=f^{\prime}(x)>0 \) (the graph is above the \( x \)-axis) in the interval \( (-1,6) \).

Step2: Determine when \( f(x) \) is decreasing

A function \( f(x) \) is decreasing when \( f^{\prime}(x)<0 \). Looking at the graph of \( y = f^{\prime}(x) \), we see that \( y = f^{\prime}(x)<0 \) (the graph is below the \( x \)-axis) in the intervals \( (-\infty,-1)\) and \( (6,\infty) \).

Answer:

a) The function \( f(x) \) is increasing on the interval \( (-1,6) \).
b) The function \( f(x) \) is decreasing on the intervals \( (-\infty,-1)\) and \( (6,\infty) \).