Final answer:
To find the interval where the function H(x) = 1/8x^3 - x^2 has a positive rate of change, we can find the critical points by taking the derivative of the function and solving for x. The interval is (0, 16/3).
Step-by-step explanation:
To determine the interval where the function H(x) = 1/8x^3 - x^2 has a positive rate of change, we can find the critical points of the function by taking its derivative. The derivative of H(x) is given by:
H'(x) = (3/8)x^2 - 2x
To find the critical points, we set H'(x) = 0 and solve for x. We have:
(3/8)x^2 - 2x = 0
This equation can be factored as:
(x)((3/8)x - 2) = 0
Setting each factor equal to zero, we find the critical points:
x = 0 and x = 16/3
Since H'(x) is a quadratic function, it opens upward, meaning that it is positive between the critical points. Therefore, the interval where H(x) has a positive rate of change is:
(0, 16/3)