Question

the graph of ( f ), the derivative of ( f ), is shown below. determine the values of ( x ), if any, at which the function ( f ) has an inflection point on the interval ( (-9,9) ).

Explanation:

Step1: Recall the definition of inflection point

An inflection point of \(y = f(x)\) occurs where the concavity of \(f(x)\) changes. This is equivalent to where the second - derivative \(f''(x)\) changes sign. Since \(f''(x)\) is the derivative of \(f'(x)\), we are looking for the points where the slope of \(y = f'(x)\) (the derivative of \(f'(x)\)) changes sign. In other words, we are looking for the local maxima or minima of \(y = f'(x)\).

Step2: Identify local maxima and minima of \(y = f'(x)\) from the graph

From the graph of \(y = f'(x)\), we can observe the following:

• The function \(y = f'(x)\) has a local maximum at \(x=-3\) (the slope of \(f'(x)\) changes from positive to negative as \(x\) increases through \(-3\)).
• The function \(y = f'(x)\) has a local minimum at \(x = 3\) (the slope of \(f'(x)\) changes from negative to positive as \(x\) increases through \(3\)).
• The function \(y = f'(x)\) has a local maximum at \(x = 6\) (the slope of \(f'(x)\) changes from positive to negative as \(x\) increases through \(6\)).

Since the slope of \(y = f'(x)\) (which is \(f''(x)\)) changes sign at \(x=-3\), \(x = 3\), and \(x = 6\) within the interval \((-9,9)\)

Answer:

\(x=-3\), \(x = 3\), \(x = 6\)