We know that $f,f'$ have exactly the same roots their gcd has (and that $f'$ also matches their multiplicity). These are $x=0$ with multiplicity $3$ and $x=\frac{5}{2}$ with multiplicity $2$.
Therefore, accounting for the increment in multiplicity,
$$f=ax^4(x- \frac{5}{2})^3$$
Given that $f(1)=3, a=-8/9$.
So our final answer should be:
$$f=-\frac{8}{9}x^4(x-\frac{5}{2})^3$$
Is this correct?
