Summarize the pertinent information cblained by appying the yrap \( f(x)=(x-4)\left(x^{2}-8 x-32\right) \) Select the correct choice below and, if necessary, fill in the answer A. The function \( f \) is concavo upward on the sutinlerval(s) (Type your answer in inlerval notation. Ise a comma lo se B. The lunction \( f \) is never concave upward. Select the correct choice helow and, if necessary, fill in the answer A. The function \( f \) is concave downward on the subinterval(s) (Type your answer in interval notation. Use a comma to sa, B. The functinn \( f \) is never concave downward Solect the correct choico below and, if necessary, fill in the answer

To analyze the concavity of the function \( f(x)=(x-4)(x^{2}-8x-32) \), we first need to find the second derivative. After computing the first derivative and then the second derivative, you can determine where the concavity changes. After using the relevant tests (the second derivative test), you may find that the function is concave upward on certain intervals, let's say \( (-\infty, 4) \cup (8, \infty) \), and concave downward on \( (4, 8) \). Be sure to calculate the critical points to verify. --- The graph of the polynomial function is fascinating! It has roots where the function intersects the x-axis, which can help you visualize the concavity. The turning points can also reveal a lot about the behavior of the function; by plotting the function, you'd see its points of inflection, where the concavity changes, making for an exciting exploration of its shape! Further exploration could lead you to study more complex functions or delve into derivative tests with real-life applications, like optimization problems. Understanding concavity can aid in a multitude of fields—from physics to economics—where predicting the behavior of systems is crucial.