Question

c) how many points of inflection does the graph of f have? justify your answer.

Explanation:

Step1: Recall the definition of a point of inflection for \( f' \)

A point of inflection on the graph of \( f' \) occurs where the second derivative of \( f \), \( f'' \), changes sign (i.e., where \( f'' \) goes from increasing to decreasing or vice - versa). Geometrically, this is where the graph of \( f' \) changes its concavity (from concave up to concave down or concave down to concave up).

Step2: Analyze the graph of \( f' \) (from the provided sketch)

Looking at the blue - sketched graph of \( f' \), we can see that there are three distinct points where the concavity of the graph changes. At each of these points, the slope of the tangent to the graph of \( f' \) (which is \( f'' \)) changes its sign. For example, if we imagine moving along the x - axis, we can identify three regions where the concavity of \( f' \) is different, and the boundaries between these regions (the points of inflection) are three in number.

Answer:

The graph of \( f' \) has three points of inflection.