Let $P(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and minimum at $x=3$.If $P(1)=6$ and $P(3)=2$, then find $P'(0)$.
Method: Since the smallest polynomial whose derivative gives 2 root would be a cubic equation. So I assumed my function to be
$$P(x) = ax^3+ bx^2+cx+d$$
Solving the condition above gives me $b=-6a$ and $c=11a$. I am stuck after this. Can anyone tell me how to proceed from here?