Let $f(x)$ be a polynomial of degree $6$ divisible by $x^3$, and having a point of extremum at $x = 2$. If $f'(x)$ is divisible by $1+x^2$, then find the value of $3f(2)/f(1)$.
What I figured out is maybe $f(x)$ should not have the terms of $x^2$, $x$ and the constant. Further, $f'(2)$ must be $0$ but that doesn't leads to the answer.
The derivative $f'$ has degree $5$, it is divisible by $x^2$, $x-2$ and $x^2+1$. Hence for some $a\not=0$, $$f'(x)=ax^2(x-2)(x^2+1)$$ which implies, since $f(0)=0$, that $$f(x)=\int_0^x f'(t)dt.$$ So what is the value of the ratio $3f(2)/f(1)$ (which does not depend on $a$)?