  A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials
$f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map
$f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a
locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We
study the accumulation set $\mathcal X$ of the critical points of the
multipliers, as $n\to\infty$. This study complements the equidistribution
result for the critical points of the multipliers that was previously obtained
by the authors. In particular, in the current paper we prove that the
accumulation set $\mathcal X$ is bounded, path connected and contains the
Mandelbrot set as a proper subset. We also provide a necessary and sufficient
condition for a parameter outside of the Mandelbrot set to be contained in the
accumulation set $\mathcal X$ and show that this condition is satisfied for an
open set of parameters. Our condition is similar in flavor to one of the
conditions that define the Mandelbrot set. As an application, we get that the
function that sends $c$ to the Hausdorff dimension of $f_c$, does not have
critical points outside of the accumulation set $\mathcal X$.
