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

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

Options:

Correct Answer: (0, 2)

Solution:

f\'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). Critical points are x = 0 and x = 3. Test intervals: f\' is positive in (0, 3) and (3, ∞).

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

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

Correct Answer: (0, 3) and (3, ∞)

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 find critical points. We can factor it as f'(x) = 4x^2(x - 3).
Step 4: Set the derivative equal to zero to find critical points: 4x^2(x - 3) = 0.
Step 5: Solve for x. This gives us two critical points: x = 0 and x = 3.
Step 6: Determine the intervals to test. The critical points divide the number line into intervals: (-∞, 0), (0, 3), and (3, ∞).
Step 7: Choose a test point from each interval to see if f'(x) is positive (increasing) or negative (decreasing).
Step 8: For the interval (-∞, 0), choose x = -1. Calculate f'(-1) = 4(-1)^2(-1 - 3) = 4(1)(-4) = -16 (negative).
Step 9: For the interval (0, 3), choose x = 1. Calculate f'(1) = 4(1)^2(1 - 3) = 4(1)(-2) = -8 (negative).
Step 10: For the interval (3, ∞), choose x = 4. Calculate f'(4) = 4(4)^2(4 - 3) = 4(16)(1) = 64 (positive).
Step 11: Identify where f'(x) is positive. The function is increasing in the interval (3, ∞).

Derivative and Critical Points – Understanding how to find the derivative of a function and identify critical points to determine intervals of increase or decrease.
Test Intervals – Using test intervals around critical points to determine where the function is increasing or decreasing.
Sign of the Derivative – Recognizing that a positive derivative indicates increasing behavior of the function.