(extremum)Let $f: \mathbb R \to \mathbb R$ be a polynomial function ...

Question

Let $f: \mathbb R \to \mathbb R$ be a polynomial function $f = a_0 + a_1x + … + a_n x^n$. Let $a_1 = a_2 = … = a_k = 0$, ($k$ less than n) and $a_{k+1} \ne 0$. The function $f$ has an extremum at the point $x=0$.

For the point $x_0$ to be an extremum of a function $f : U(x_0) \to \mathbb R$, defined in a neighborhood of $x_0$, it is necessary that one of the two conditions is satisfied:

1. $f$ is not derivable in $x_0$

or

2. $f_0(x_0) = 0$.

$f'(0) = a_1 + ... + na_n x^{n-1} = a_1 = 0$

since f'(0) = 0 we cannot guarantee the existence or not of the extreme.