The force associated with this potential is
$\begin{align*}F = - 2 b x\end{align*}$
So, the spring constant is $k = 2 b$ and the angular frequency of the oscillations is:
$\begin{align*}\omega = \sqrt{\frac{2b}{m}}\end{align*}$
This depends only on $b$ and $m$ , hence answer (C) is correct.

Solution to 1986 Problem 60