If inflexion, then no. Consider y = x^5, which does have its third derivative = 0 at x = 0. And futhermore, consider
y = x^4
dy/dx = 4x^3
d^2y/dx^2 = 12x^2
d^3y/dx^3 = 24x
x = 1 ==> d^3y/dx^3 = 24, so according to your theory, there is a point of inflexion there.
x = 2 ==> d^3y/dx^3 = 48, so according to your theory, there is a point of inflexion there.
etc