Question

use the given graph of ( y = f(x) ) to find the intervals on which ( f(x)>0 ), the intervals on which ( f(x)<0 ), and the values of ( x ) for which ( f(x)=0 ). sketch a possible graph of ( y = f(x) ).
on what subinterval(s) is ( f(x)>0 )? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. ( (-infty,-2),(4,9) )
(type your answer using interval notation. use a comma to separate answers as needed.)
b. there is no solution.
on what subinterval(s) is ( f(x)<0 )? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a.
(type your answer using interval notation. use a comma to separate answers as needed.)
b. there is no solution.

Explanation:

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

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

Step2: Analyze the graph of \(y = f(x)\) for \(f^{\prime}(x)<0\)

Since \(f^{\prime}(x)>0\) on \((-\infty,- 2)\cup(4,9)\) (given as correct for \(f^{\prime}(x)>0\)), the function \(y = f(x)\) is decreasing on the intervals where \(f^{\prime}(x)<0\). The function \(y = f(x)\) is decreasing on \((-2,4)\)

Answer:

For \(f^{\prime}(x)<0\): A. \((-2,4)\)