What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)

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Question: What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)

Options:

Correct Answer: 1

Exam Year: 2022

Solution:

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

What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)

What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)

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 the equation 3x^2 - 3 = 0. This simplifies to x^2 = 1.
Step 5: Take the square root of both sides to find x = ±1.
Step 6: To determine if these points are local maxima or minima, evaluate the second derivative f''(x) = 6x.
Step 7: Check the second derivative at x = 1: f''(1) = 6(1) = 6 (positive, so it's a local minimum).
Step 8: Check the second derivative at x = -1: f''(-1) = 6(-1) = -6 (negative, so it's a local maximum).
Step 9: Conclude that the value of x where f(x) has a local maximum is x = -1.

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