Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.

Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.

Correct Answer: (2, 2)

Step 1: Start with the function f(x) = x^4 - 4x^3 + 6.
Step 2: Find the first derivative f'(x) to determine the slope of the function.
Step 3: Calculate f'(x) = 4x^3 - 12x^2.
Step 4: Find the second derivative f''(x) to determine the concavity of the function.
Step 5: Calculate f''(x) = 12x^2 - 24x.
Step 6: Set the second derivative equal to zero: 12x^2 - 24x = 0.
Step 7: Factor the equation: 12x(x - 2) = 0.
Step 8: Solve for x: This gives x = 0 or x = 2.
Step 9: To find the point of inflection, we need to find the corresponding y-value for x = 2.
Step 10: Calculate f(2) = (2)^4 - 4(2)^3 + 6 = 16 - 32 + 6 = -10.
Step 11: The point of inflection is at (2, -10).

Second Derivative Test – Understanding how to find points of inflection using 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 identified points to determine the coordinates of the inflection point.