Is there a relationship between the stationary points and the inflection point of a cubic polynomial function? Determine the stationary points, A and B, and point of inflection, G, for each of the following cubic polynomials.
(a) $y=x^3 -3x^2 -9x+7$ 
(b) $y=x^3 -12x^2 +21x-14$
(c) $y=x^3 +9x^2 -12$
Is there any common relationship between A, B and G?

For any cubic, by translating the independent variable, you can make the quadratic term disappear and obtain a form
$$x^3+px+q.$$
You can also translate vertically and reduce to
$$x^3+px.$$
Then by rescaling the variable and the function,
$$\frac{(sx)^3+psx}{s^3}=x^3+\frac p{s^2}x=x^3\pm \,x$$ by setting $s^2=|p|$. And if $p=0$, we simply have $x^3$.
These transformation are affine and preserve ratios, and in particular the relative positions of the extrema and the inflection.
Without further computation, as the function is odd the single inflection must lie midway of the two extrema.

We can in fact say that, as regards the remarkable points, there are essentially three cubics: $x^3,x^3-x,x^3+x$.

(If the cubic coefficient was negative, the graph must be flipped and one may distinguish six cases, $\pm x^3,\pm x^3\pm x$.)
