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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.
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Practice Questions
1 question
Q1
Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.
(1, 4)
(2, 3)
(3, 0)
(0, 0)
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f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).
Questions & Step-by-step Solutions
1 item
Q
Q: Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.
Solution:
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).
Steps: 10
Show Steps
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).
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