f(x) = ax^3 + bx^2 + cx + d f'(x) = 3ax^2 + 2bx + c For the function to be either increasing or decreasing on the intervel (-inf, +inf), f'(x) = 0 should not have more than one root, because two roots mean there is a maxima and a minima, one root means there is one point of inflection, and no root means there is no mamima, minima, or point of inflection. So, Required condition is, discriminant of f'(x) <= 0; (0: one root, <0: no root) or, (2b)^2 - 4(3a).c <= 0 or, b^2 <= 6ac