1. **Problem Statement:** We are given a curve $\hat{f}(x)$ and asked to identify which of the statements (a) to (d) about the function $f$ are correct, except one. 2. **Understanding the statements:** - (a) At $x = -5$ there is a local maximum of $f$. - (b) $f''(1) > 0$ (the second derivative at $x=1$ is positive). - (c) At $x = 7$ there is a local minimum of $f$. - (d) The function $f$ is decreasing on the interval $(-5,1)$. 3. **Analyzing the graph and statements:** - At $x = -5$, the curve $\hat{f}$ is at a peak, indicating a local maximum for $f$ (statement a is correct). - At $x = 1$, the curve changes concavity. Since $f''(1) > 0$ means the curve is concave up at $x=1$, this matches the graph's shape (statement b is correct). - At $x = 7$, the curve reaches a local minimum (statement c is correct). - On the interval $(-5,1)$, the function $f$ is actually increasing from the local maximum at $-5$ down to the minimum near $0$ and then rising again to $1$, so it is not decreasing throughout this interval (statement d is incorrect). 4. **Conclusion:** The incorrect statement is (d). **Final answer:** The statement that is NOT correct is (d) The function $f$ is decreasing on $(-5,1)$.