The function f(x) = x^4 - 4x^3 + 4x^2 has a point of inflection at x =

Core Concept

Find the second derivative: f'(x) = 4x^3 - 12x^2 + 8x, and f''(x) = 12x^2 - 24x + 8. Set f''(x) = 0: 12x^2 - 24x + 8 = 0, which simplifies to 3x^2 - 6x + 2 = 0. Using the quadratic formula: x = (6 ± √(36 - 24))/6 = (6 ± √12)/6 = (6 ± 2√3)/6 = (3 ± √3)/3. However, we can also check the sign changes of f''(x). At x = 0, f''(0) = 8 > 0. At x = 1, f''(1) = 12 - 24 + 8 = -4 < 0. At x = 2, f''(2) = 48 - 48 + 8 = 8 > 0. Since f''(x) changes sign at both x = 0 and x = 1, these are points of inflection.