The diagram shows the graph of $y = f(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2016 - Paper 1

The graph of $y = f(x)$ shows a maximum point at $x = 1$, where $f'(1) = 0$, indicating that the derivative is zero at this point. The second derivative, $f''(1)$, will be negative since the graph is concave down at $x = 1$.

Thus, we can analyze the provided statements:

• (A) $f''(1) < f'(1) < 1 < f(1)$: This is incorrect, as $f'(1) = 0$ and $f''(1)$ is negative.
• (B) $f''(1) < f'(1) < f(1) < 1$: This implies that $f'(1) < f(1)$, which is wrong because $f(1) = 1$ and $f'(1) = 0$.
• (C) $f(1) < 1 < f'(1) < f''(1)$: This is incorrect, as we have $f(1) = 1$.
• (D) $f'(1) < f(1) < 1 < f''(1)$: This is also incorrect as $f'(1) = 0$, $f(1) = 1$, and $f''(1) < 0$.

Given the analysis, the correct statement based on the graph is A: $f''(1) < f'(1) < 1 < f(1)$.