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 changes. Concavity is related to the second - derivative \(f''(x)\). Since \(f''(x)\) is the derivative of \(f'(x)\), an inflection point of \(f(x)\) occurs where \(f'(x)\) has a local maximum or minimum (because at these points, the sign of \(f''(x)\) changes).

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

Looking at the graph of \(y = f'(x)\), we find the \(x\) - values of the local maxima and minima.
From the graph, the local maxima of \(y = f'(x)\) occur at \(x=-4\) and \(x = 1\), and the local minima occur at \(x = 4\)

Answer:

\(x=-4,1,4\)