Final answer:
The function f(x) = x³ + 3x² - 2 is decreasing over the interval (-2, 0), determined by analyzing its first derivative and solving the inequality 3x² + 6x < 0.
Step-by-step explanation:
To determine over which interval the function f(x) = x³ + 3x² - 2 is decreasing, we need to analyze its first derivative, f'(x). A function is decreasing where its derivative is negative. Let's find the first derivative of f(x):
f'(x) = 3x² + 6x
To find where f'(x) is negative, we solve the inequality 3x² + 6x < 0. Factoring out a 3x from f'(x):
3x(x + 2) < 0
This suggests our critical points are x = 0 and x = -2. A sign chart for f'(x) around these critical points shows that the derivative is negative (and thus f(x) is decreasing) in the interval (-2, 0). Therefore, the function f(x) is decreasing over the interval (-2, 0).