I will solve this problem without taking any derivatives.

x^3 - 6x^2 + 12x + 6 = x^3 - 6x^2 + 12x - 8 + 14 = (x - 2)^3 + 14

The function is always increasing and has an inflection at x = 2, or (2, 14)

It is concave downward for x < 2 and concave upward for x > 2