Final answer:
The function h(x) = 3√x is increasing on the interval (0, ∞), as it is a cubic root function that increases with increasing x. It is not defined for negative values of x, thus it cannot be increasing or decreasing on the interval (-∞, 0).
Step-by-step explanation:
The student asks about the behavior of the function h(x) = 3√x over different intervals. To determine if the function is increasing or decreasing, we need to look at the derivative h'(x). For the cubic root function, it is well known that the function increases as x increases; however, it is important to note that h(x) is only defined for x >= 0, as the real cubic root of a negative number is not defined in the real number system.
Therefore, the function h(x) does not exist on the interval (-∞, 0), making statements 1 and 2 incorrect. On the interval (0, ∞), the cubic root function increases as x increases, which means the slope of the graph is positive, and thus, h(x) is increasing on this interval. This makes statement 4 the correct choice: The function is increasing on the interval (0, ∞).