Step 7: Solve for x to find critical points: x = 1 and x = 2.
Step 8: Identify the intervals to test: (-∞, 1), (1, 2), and (2, ∞).
Step 9: Choose a test point in each interval and plug it into f'(x) to see if it's positive (increasing) or negative (decreasing).
Step 10: For the interval (-∞, 1), test x = 0: f'(0) = 12 (positive, so increasing).
Step 11: For the interval (1, 2), test x = 1.5: f'(1.5) = -1.5 (negative, so decreasing).
Step 12: For the interval (2, ∞), test x = 3: f'(3) = 12 (positive, so increasing).
Step 13: Combine the results: The function is increasing on the intervals (0, 1) and (2, ∞).
Derivative and Critical Points – Understanding how to find the derivative of a function to identify critical points where the function may change from increasing to decreasing or vice versa.
Test Intervals – Using test intervals around critical points to determine where the function is increasing or decreasing.
Function Behavior – Analyzing the behavior of polynomial functions to determine intervals of increase and decrease.
