Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)

Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)

Step 1: Start with the function f(x) = x^4 - 4x^3 + 6.
Step 2: Find the first derivative f'(x) to understand 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 us x = 0 and x = 2.
Step 9: To confirm these are points of inflection, check the sign of f''(x) around these points.
Step 10: Choose a test point, like x = 1, and calculate f''(1) to see if it changes sign.
Step 11: Calculate f''(1) = 12(1)^2 - 24(1) = 12 - 24 = -12 (negative).
Step 12: Check f''(3) = 12(3)^2 - 24(3) = 108 - 72 = 36 (positive).
Step 13: Since f''(x) changes from negative to positive at x = 2, this confirms a point of inflection.

Second Derivative Test – The second derivative is used to determine concavity and identify points of inflection.
Finding Critical Points – Setting the second derivative equal to zero to find potential points of inflection.
Verification of Inflection Points – Checking the sign of the second derivative around the critical points to confirm they are indeed points of inflection.