Determine the intervals where f(x) = x^3 - 3x is increasing. (2021)

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Question: Determine the intervals where f(x) = x^3 - 3x is increasing. (2021)

Options:

Correct Answer: (1, ∞)

Exam Year: 2021

Solution:

f\'(x) = 3x^2 - 3. Setting f\'(x) = 0 gives x = -1, 1. f\'(x) > 0 for x > 1.

Determine the intervals where f(x) = x^3 - 3x is increasing. (2021)

Determine the intervals where f(x) = x^3 - 3x is increasing. (2021)

Step 1: Write down the function f(x) = x^3 - 3x.
Step 2: Find the derivative of the function, which is f'(x) = 3x^2 - 3.
Step 3: Set the derivative equal to zero to find critical points: 3x^2 - 3 = 0.
Step 4: Solve for x: 3x^2 = 3, so x^2 = 1, which gives x = -1 and x = 1.
Step 5: Identify the intervals to test: (-∞, -1), (-1, 1), and (1, ∞).
Step 6: Choose a test point in each interval: for (-∞, -1) use x = -2, for (-1, 1) use x = 0, and for (1, ∞) use x = 2.
Step 7: Calculate f'(-2), f'(0), and f'(2): f'(-2) = 3(-2)^2 - 3 = 9 > 0, f'(0) = 3(0)^2 - 3 = -3 < 0, f'(2) = 3(2)^2 - 3 = 9 > 0.
Step 8: Determine where f'(x) is positive: f'(x) > 0 in the intervals (-∞, -1) and (1, ∞).
Step 9: Conclude that f(x) is increasing on the intervals (-∞, -1) and (1, ∞).

Derivative and Critical Points – Understanding how to find the derivative of a function and identify critical points where the function's behavior changes.
Increasing and Decreasing Intervals – Determining where the function is increasing or decreasing based on the sign of the derivative.