If f(x) = x^3 - 6x^2 + 9x, find the inflection point. (2023)

If f(x) = x^3 - 6x^2 + 9x, find the inflection point. (2023)

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the second derivative of the function, f''(x).
Step 3: To find f''(x), first find the first derivative f'(x) = 3x^2 - 12x + 9.
Step 4: Now, find the second derivative f''(x) by differentiating f'(x): f''(x) = 6x - 12.
Step 5: Set the second derivative equal to zero to find potential inflection points: 6x - 12 = 0.
Step 6: Solve for x: 6x = 12, so x = 2.
Step 7: To find the inflection point, calculate f(2) by substituting x = 2 into the original function: f(2) = 2^3 - 6(2^2) + 9(2).
Step 8: Calculate f(2): f(2) = 8 - 24 + 18 = 2.
Step 9: The inflection point is (2, f(2)), which is (2, 2).

Second Derivative Test – The inflection point is found by determining where the second derivative changes sign, indicating a change in concavity.
Finding Inflection Points – An inflection point occurs where the second derivative is zero or undefined, and the concavity changes.