How to find the value of a parameter such that the map has a period-doubling bifurcation?

Question

For example:

$f(x)=x_{n+1}=\mu+x_n^2$.

Is it when $|f'(x^*)|=1$, where $x^*$ is a fixed point of the system? In this case, $\mu=1/4$?

Also how to determine whether it is supercritical or subcritical? Is it supercritical when $|f^{2'}(x^{**})|<1$ and $\mu=1/4$, where $x^{**}$ is the fixed point of 2-period?