Surnmarize the pertinent information obtained by \( f(x)=2 x(x-3)^{3} \) On what interval(s) is \( f \) concave upward or conca (Type your answer in interval notation. Use integ A. \( f \) is concave upward on \( \square \) and conceve B. \( f \) is never concave downward, \( f \) is concav C. \( f \) is never concave upward; \( f \) is concave \( d \)

To determine the intervals of concavity for the function \( f(x) = 2x(x-3)^{3} \), we start by finding the second derivative \( f''(x) \). 1. Calculate the first derivative \( f'(x) \) and then the second derivative \( f''(x) \). 2. Identify the points where \( f''(x) = 0 \) or is undefined, as these points will help us find critical points for concavity. 3. Test intervals around these points to see where \( f''(x) \) is positive (concave upward) or negative (concave downward). Upon analysis, you would find that: A. \( f \) is concave upward on \( (0, 3) \) and concave downward on \( (3, \infty) \). B. \( f \) is never concave downward; \( f \) is concave upward on the stated interval. C. \( f \) is never concave upward; \( f \) is concave down after \( x=3 \).