Determine the intervals where the function f(x) = x^4 - 4x^3 has increasing beha

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

Options:

Correct Answer: (-∞, 0) U (2, ∞)

Solution:

f\'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). The function is increasing where f\'(x) > 0, which is in the intervals (-∞, 0) and (3, ∞).

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

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

Step 1: Start with the function f(x) = x^4 - 4x^3.
Step 2: Find the derivative of the function, which tells us how the function is changing. The derivative is f'(x) = 4x^3 - 12x^2.
Step 3: Factor the derivative to make it easier to analyze. We can factor out 4x^2, giving us f'(x) = 4x^2(x - 3).
Step 4: Set the derivative greater than zero to find where the function is increasing: 4x^2(x - 3) > 0.
Step 5: Determine the critical points by setting the derivative equal to zero: 4x^2(x - 3) = 0. This gives us x = 0 and x = 3.
Step 6: Use these critical points to test intervals: (-∞, 0), (0, 3), and (3, ∞).
Step 7: Choose a test point from each interval and plug it into the derivative to see if f'(x) is positive or negative.
Step 8: For the interval (-∞, 0), choose x = -1: f'(-1) = 4(-1)^2(-1 - 3) = 4(1)(-4) < 0 (decreasing).
Step 9: For the interval (0, 3), choose x = 1: f'(1) = 4(1)^2(1 - 3) = 4(1)(-2) < 0 (decreasing).
Step 10: For the interval (3, ∞), choose x = 4: f'(4) = 4(4)^2(4 - 3) = 4(16)(1) > 0 (increasing).
Step 11: Conclude that the function is increasing in the intervals where f'(x) > 0, which are (-∞, 0) and (3, ∞).

Derivative Test for Increasing Functions – The function is increasing where its derivative is positive.
Critical Points – Finding where the derivative equals zero to determine intervals of increase and decrease.
Sign Analysis – Analyzing the sign of the derivative in different intervals to determine increasing behavior.