Question

given the graph of the function ( f ) below, determine all intervals on the open interval ( (-9,9) ) where ( f(x) geq 0 ) and ( f(x) leq 0 ).

Explanation:

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

When \(f^{\prime}(x)\geq0\), the function \(f(x)\) is non - decreasing. When \(f(x)\leq0\), the function \(f(x)\) is non - increasing.

Step2: Analyze the graph

Looking at the graph of \(y = f(x)\):

• For the interval \((- 9,-7)\), the function \(f(x)\) is non - decreasing (\(f^{\prime}(x)\geq0\)) and \(f(x)\leq0\) (the graph lies below the \(x\) - axis).
• For the interval \((-5,-3)\), the function \(f(x)\) is non - decreasing (\(f^{\prime}(x)\geq0\)) and \(f(x)\leq0\) (the graph lies below the \(x\) - axis).

Answer:

The intervals are \((-9,-7)\) and \((-5,-3)\)