Find a polynomial $f \in \mathbb{Q}[x]$ with the least possible degree s.t. $\gcd(f,f')= x^5-5x^4+\frac{25}{4}x^3, f(1)=3$

Question

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?