Final answer:
None of the statements A, B, C, or D are exclusively true. A polynomial p(x) modeling the volume of a box with x intercepts at -2, 10, and 14 would be of the form p(x) = k(x + 2)(x - 10)(x - 14) with k > 0. The maximum volume would not occur at an x intercept, making C incorrect, and there's insufficient data to determine the truth of D.
Step-by-step explanation:
The given polynomial equation of degree three, p(x), has x intercepts at -2, 10, and 14. This means p(x) will be zero at these x-values. Therefore, we can write the polynomial as p(x) = k(x + 2)(x - 10)(x - 14), where k is a leading coefficient. If the graph of p(x) represents the volume of a box, then k should be positive because volume cannot be negative. This rules out option B because a negative sign would render k negative, leading to negative volume.
Statement A suggests a function that is incorrect because it contains (x - 2) instead of (x + 2). Turning our attention to C and D, the maximum volume corresponds to the highest point (vertex) of the graph of the polynomial when the leading coefficient is positive. The highest point cannot be at an x-intercept; hence, C is incorrect. Without specific values to calculate the maximum volume, we cannot determine the truth of statement D.
Therefore, none of the provided statements are exclusively true. If required to choose, depending on the sign of k (positive for volume), a modified version of A or B could be valid, but with a positive coefficient making the function p(x) = k(x + 2)(x - 10)(x - 14) where k > 0.