Find the value of x for which the function f(x) = x^3 - 6x^2 + 9x has a point of

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

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

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the first derivative f'(x) to understand 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 by adding 12 to both sides: 6x = 12.
Step 8: Divide both sides by 6 to find x: x = 2.
Step 9: The value of x for which the function has a point of inflection is x = 2.

Inflection Points – Inflection points occur where the second derivative of a function changes sign, indicating a change in the concavity of the function.
Second Derivative Test – The second derivative test is used to determine the concavity of a function and to find points of inflection by setting the second derivative equal to zero.