I have a function $f(x) = \prod_{i=1}^n (x-\alpha_{i})^{e_{i}}$. $f \in F[x]$ and $f' \in F[x]$. $f$ is monic. This means that there is a field extension $L$ of $F$ such that $f(x)$ exists in that format with distinct integers $\alpha_0,...,\alpha_n \in L$ and $e_i \ge 1$.
Now, I am supposed to show that $e_i \ge 2$ if and only if $f'(\alpha_i) = 0$. I am very lost as to how to approach this problem. Should I just attempt to take the derivative of $f$ and see what happens? Help would be appreciated. Thanks.
$f^\prime(x)= \sum_i e_i(x-\alpha_i)^{e_i-1}\prod_{j \neq i} (x-\alpha_j)^{e_j}$