special polynomials

I'm searching for a polynomial $f$ of degree 4 with the following property: $f$ and all its derivatives have the maximum number of integer roots.

Concretely formulated:

$$\begin{eqnarray*} f(x) & = & (x-a)(x-b)(x-c)(x-d) \\ f'(x) & = & 4(x-e)(x-f)(x-g) \\ f''(x) & = & 12(x-h)(x-i) \\ f'''(x) &= & 24(x-j) \\ \end{eqnarray*}$$

should be satisfied simultaneously with distinct integers $a,b,c,d$, distinct integers $e,f,g$, disctinct integers $h,i$ and an integer $j$.

My conjecture is that there is no such polynomial. For degree 3, there are solutions. Can anyone either prove this or find a counterexample?