Step 7: Solve for x to find the critical points: x = 1 and x = 3.
Step 8: Use the critical points to test intervals: (-∞, 1), (1, 3), and (3, ∞).
Step 9: Choose a test point in each interval and plug it into f'(x) to see if the derivative is positive (increasing) or negative (decreasing).
Step 10: For the interval (-∞, 1), choose x = 0: f'(0) = 9 (positive, so increasing).
Step 11: For the interval (1, 3), choose x = 2: f'(2) = 3(2^2) - 12(2) + 9 = 3 (positive, so increasing).
Step 12: For the interval (3, ∞), choose x = 4: f'(4) = 3(4^2) - 12(4) + 9 = 9 (positive, so increasing).
Step 13: Conclude that the function is increasing on the intervals (1, 3) and (3, ∞).
Derivative and Critical Points – Understanding how to find the derivative of a function to determine critical points and analyze intervals of increase and decrease.
Increasing and Decreasing Intervals – Identifying where the function is increasing based on the sign of the derivative.
