Question

given the graph of the function ( f ) below, determine all intervals on the open interval ( (-9,9) ) where ( f^{prime}(x) leq 0 ).

Explanation:

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

If \(f^{\prime}(x)\leq0\), the function \(f(x)\) is non - increasing. A function \(y = f(x)\) is non - increasing when, as \(x\) increases, \(y\) does not increase. Visually, the slope of the tangent line to the graph of \(y = f(x)\) is non - positive.

Step2: Analyze the graph

Looking at the graph of \(y = f(x)\) in the open interval \((-9,9)\):

• The function \(f(x)\) is non - increasing in the intervals \((- 9,-6)\) and \((7,9)\)

Answer:

\((-9,-6)\cup(7,9)\)