Final answer:
To determine the intervals of increase and decrease for the function f(x) = 2x³ - 6x² + 9, we find the derivative f'(x) = 6x² - 12x, set it to zero to find critical points which are x = 0 and x = 2. By testing intervals around these points, we find that f(x) increases on the intervals (-∞, 0) and (2, ∞), and decreases on the interval (0, 2).
Step-by-step explanation:
To find the intervals of increase and decrease for the function f(x) = 2x³ - 6x² + 9, we need to determine where the derivative of the function changes sign. The derivative f'(x) represents the slope of the function at any given point, and where this derivative is positive, the function is increasing, and where it is negative, the function is decreasing.
First, we find the derivative:
f'(x) = 6x² - 12x.
Next, we find the critical points by setting f'(x) = 0:
6x² - 12x = 0
x(6x - 12) = 0
x = 0 or x = 2.
These critical points divide the number line into intervals that we can test to determine where f(x) is increasing or decreasing:
- Interval (1): x < 0
- Interval (2): 0 < x < 2
- Interval (3): x > 2
To determine the sign of f'(x) on these intervals, pick test points and evaluate f'(x) at these points:
- For x = -1 (Interval 1), f'(-1) = 6(-1)² - 12(-1) = 6 + 12 = 18 (positive)
- For x = 1 (Interval 2), f'(1) = 6(1)² - 12(1) = 6 - 12 = -6 (negative)
- For x = 3 (Interval 3), f'(3) = 6(3)² - 12(3) = 54 - 36 = 18 (positive)
Therefore, f(x) is:
- Increasing on the interval (-∞, 0) and (2, ∞)
- Decreasing on the interval (0, 2)