Final answer:
To determine over which interval the function h(x) = 1/8x^3 - x^2 has a positive rate of change, we need to find the intervals where the derivative of the function is positive. The function h(x) has a positive rate of change over the interval (-∞, 0) U (8/3, ∞).
Step-by-step explanation:
To determine over which interval the function h(x) = 1/8x^3 - x^2 has a positive rate of change, we need to find the intervals where the derivative of the function is positive. Let's find the derivative of h(x):
h'(x) = (3/8)x^2 - 2x
To find where h'(x) > 0, we need to solve the inequality:
(3/8)x^2 - 2x > 0
Simplifying and factoring, we get:
(x - 0)(x - 8/3) > 0
This inequality is true when either both factors are positive or both factors are negative:
x - 0 > 0 and x - 8/3 > 0, or x - 0 < 0 and x - 8/3 < 0
Simplifying, we get:
x > 0 and x > 8/3, or x < 0 and x < 8/3
The solution set is:
x > 8/3 or x < 0
Therefore, the function h(x) has a positive rate of change over the interval (-∞, 0) U (8/3, ∞).