Question

the graph of ( f ), the derivative of the function ( f ), is shown below. determine all intervals on which ( f(x) ) is concave down on the open interval ( (-9,9) ).

Explanation:

Step1: Recall the concavity rule

A function \(y = f(x)\) is concave down when \(f''(x)<0\). Since \(f''(x)\) is the derivative of \(f'(x)\), we need to find where the slope of \(y = f'(x)\) is negative.

Step2: Analyze the graph of \(y = f'(x)\)

Looking at the graph of \(y = f'(x)\), we observe the intervals where the function \(y = f'(x)\) is decreasing. A decreasing function \(y = f'(x)\) implies \(f''(x)<0\).

From the graph, \(y = f'(x)\) is decreasing on the intervals \((-9,-7)\) and \((0,4)\)

Answer:

The function \(f(x)\) is concave down on the intervals \((-9, - 7)\) and \((0,4)\)