Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)

Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)

Step 1: Start with the function f(x) = x^4 - 4x^3.
Step 2: Find the derivative of the function, f'(x). This tells us how the function is changing.
Step 3: Calculate the derivative: f'(x) = 4x^3 - 12x^2.
Step 4: Factor the derivative: f'(x) = 4x^2(x - 3).
Step 5: Set the derivative greater than zero to find where the function is increasing: 4x^2(x - 3) > 0.
Step 6: Identify the critical points by setting the derivative equal to zero: 4x^2(x - 3) = 0. This gives us x = 0 and x = 3.
Step 7: Test intervals around the critical points (choose test points in the intervals (-∞, 0), (0, 3), and (3, ∞)).
Step 8: For the interval (0, 3), choose a test point like x = 1. Calculate f'(1) = 4(1)^2(1 - 3) = 4(1)(-2) < 0, so it's decreasing.
Step 9: For the interval (3, ∞), choose a test point like x = 4. Calculate f'(4) = 4(4)^2(4 - 3) = 4(16)(1) > 0, so it's increasing.
Step 10: Conclude that f(x) is increasing on the interval (3, ∞).

Derivative Test for Increasing Functions – The question tests the understanding of how to find intervals of increasing behavior for a function by analyzing its first derivative.
Critical Points – Identifying critical points where the derivative is zero or undefined is essential for determining intervals of increase or decrease.
Sign Analysis – The ability to perform sign analysis on the derivative to determine where the function is increasing or decreasing.