Using the sign chart for \( f^{\prime \prime}(x)=-3(x+8)^{3}(x+4)^{2} \), and only that sign chart, what conclusions can be drawn about the graph of \( f \) on the interval \( x<-8 \) ? Answer \( \begin{array}{ll}\text { it is positive } & \text { it is negative } \\ \text { it is increasing } & \text { it is decreasing } \\ \text { is concave up } & \text { it is concave down } \\ \text { none of the above } & \text { Submit Answer }\end{array} \)

To analyze the behavior of the function \( f \) on the interval \( x < -8 \) using the second derivative \( f''(x) = -3(x+8)^3(x+4)^2 \), let's follow these steps: 1. **Determine the Sign of \( f''(x) \) for \( x < -8 \):** Choose a test point in the interval \( x < -8 \), for example, \( x = -9 \): \[ f''(-9) = -3(-9+8)^3(-9+4)^2 = -3(-1)^3(-5)^2 = -3(-1)(25) = 75 \] Since \( f''(-9) > 0 \), \( f''(x) \) is positive on \( x < -8 \). 2. **Interpret the Sign of \( f''(x) \):** - **Concavity:** A positive second derivative (\( f''(x) > 0 \)) indicates that the graph of \( f \) is **concave up** on this interval. - **Behavior of \( f'(x) \):** While \( f''(x) > 0 \) implies that the first derivative \( f'(x) \) is increasing, this information alone does not directly tell us whether \( f(x) \) itself is increasing or decreasing without additional data about \( f'(x) \). 3. **Conclusion:** Based solely on the sign chart for \( f''(x) \) and the fact that \( f''(x) > 0 \) for \( x < -8 \), the primary conclusion we can draw is about the concavity of \( f \). **Answer:** The graph of \( f \) is **concave up** on the interval \( x < -8 \).