For which values of $a$, $f(x)=1+a(a-1)(a+2)x+ax^2+x^3+o(x^3)$ has a minimum in $x=0$?

Question

I would like to find for which values of $a$, $f(x)=1+a(a-1)(a+2)x+ax^2+x^3+o(x^3)$ has a minimum in $x=0$. At first glance, I would take the derivative of $f(x)$ and do $f'(x)=0$. Then, check where there are minumum and maximum values. The problem comes with the parameter $a$. Can anyone help me?

Answer 1

$0=f'(0)=a(a-1)(a+2)\implies a=0,1$ or $-2$.

Meanwhile, $0\le f''(0)=2a\implies a\ge0$.

Hence $a=0,1$.