Final answer:
Upon calculating the second derivative of the function f(x) = ∛(4x), we determine there is no interval where the function is concave up and no inflection points, as the sign of the second derivative does not change from negative, which indicates the function is concave down for all x>0.
Step-by-step explanation:
To determine the intervals of concavity and inflection points for the function f(x) = ∛(4x), we need to find the second derivative of the function, since the concavity is determined by the sign of the second derivative. We start by finding the first derivative f'(x) = ¼ * 4x-¾ or f'(x) = 1/(3∛(4x2)). Then, we find the second derivative f''(x) = -2/9 * 4x-⅔ or f''(x) = -2/(9∛(4x5)). As x is positive, the second derivative is negative, thus the function is concave down. As x is negative, the second derivative does not exist because the cube root of a negative number is not real in the context of this function. Therefore, the interval of concavity is:
- Concave Up: None
- Concave Down: (0, ∞)
However, there seems to be a contradiction in the options provided and a misunderstanding of the function's behavior, as the concavity given in the options does not match the calculated one. Specifically, there is no interval where the function is concave up, and no inflection point as the sign of the second derivative does not change.