The diagram shows the graph of the curve $y = f(x)$ - HSC - SSCE Mathematics Extension 2 - Question 8 - 2018 - Paper 1

To determine where the concavity of the curve $y = F(x)$ changes, we need to analyze the second derivative of $F(x)$, which tells us about the concavity. The concavity of a function changes at points where the second derivative equals zero or is undefined.

1. Find the First Derivative: Using the Fundamental Theorem of Calculus, the first derivative of $F(x)$ is given by: $F'(x) = f(x)$

2. Find the Second Derivative: The second derivative, which gives information about concavity, is: $F''(x) = f'(x)$

3. Find Critical Points: To find the points where concavity changes, we need to find where $f'(x) = 0$. From the graph, we identify the critical points that correspond to where the slope of the tangent line (represented by $f'(x)$) is zero.

4. Analyze the Graph: From the graph provided:

   • The curve appears to change concavity at the points labelled a, c, and d.

Thus, the answer is B. a, c.