For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.

For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.

Correct Answer: (1.5, f(1.5))

Step 1: Start with the function f(x) = 2x^3 - 9x^2 + 12x.
Step 2: Find the first derivative f'(x) to determine the slope of the function.
Step 3: Find the second derivative f''(x) to determine the concavity of the function.
Step 4: Set the second derivative f''(x) equal to 0 to find potential inflection points.
Step 5: Solve the equation from Step 4 to find the x-value of the inflection point.
Step 6: Substitute the x-value found in Step 5 back into the original function f(x) to find the corresponding y-value.
Step 7: Combine the x and y values to write the inflection point as (x, y).

Second Derivative Test – The inflection point is found by setting the second derivative equal to zero and solving for x.
Function Evaluation – After finding the x-coordinate of the inflection point, the corresponding y-coordinate must be calculated by substituting x back into the original function.