What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023)

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Question: What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023) 2023

Options:

Correct Answer: (1, 4)

Exam Year: 2023

Solution:

To find inflection points, set f\'\'(x) = 0. f\'\'(x) = 6x - 12 = 0, x = 2. f(2) = 4.

What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023) 2023

What is the point of inflection for the function f(x) = x^3 - 6x^2 + 9x? (2023) 2023

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the first derivative f'(x) to determine the slope of the function.
Step 3: Calculate f'(x) = 3x^2 - 12x + 9.
Step 4: Find the second derivative f''(x) to determine the concavity of the function.
Step 5: Calculate f''(x) = 6x - 12.
Step 6: Set the second derivative equal to zero: 6x - 12 = 0.
Step 7: Solve for x: 6x = 12, so x = 2.
Step 8: To find the y-coordinate of the inflection point, substitute x = 2 back into the original function: f(2) = 2^3 - 6(2^2) + 9(2).
Step 9: Calculate f(2) = 8 - 24 + 18 = 2.
Step 10: The point of inflection is (2, 2).

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