Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.

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Question: Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.

Options:

Correct Answer: (3, 0)

Solution:

f\'\'(x) = 6x - 12. Setting f\'\'(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).

Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.

Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.

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: Find the y-coordinate of the point of inflection by calculating f(2).
Step 9: Calculate f(2) = 2^3 - 6(2^2) + 9(2) = 8 - 24 + 18 = 2.
Step 10: The point of inflection is (2, 2).

Second Derivative Test – Understanding how to find points of inflection by analyzing the second derivative of a function.
Critical Points – Identifying where the second derivative equals zero to find potential points of inflection.
Function Evaluation – Calculating the function value at the point of inflection to determine the coordinates.