= Solution
The fixed-point equation is $x[ax^2+bx+\mu-1]=0$. Thus
$$
\boxed{x=0,\qquad x_\pm=\frac{-b\pm\sqrt{b^2+4a(1-\mu)}}{2a}}
$$
when the discriminant is nonnegative. At zero the multiplier is $F'(0)=\mu$, so it is linearly stable for $-1<\mu<1$ and unstable for $|\mu|>1$. At a nonzero fixed point, eliminate $\mu$ to obtain the branch and its multiplier:
$$
\mu=1-bx-ax^2,\qquad M=1+bx+2ax^2.
$$
The branch is stable precisely when $-2<x(b+2ax)<0$. These formulas specify stability throughout the diagram, including any extra flip points outside the region requested.
If $b=0$ and $a>0$, the two nonzero branches $x=\pm\sqrt{(1-\mu)/a}$ exist for $\mu<1$ and have multiplier $3-2\mu>1$. Thus $\mu=1$ is a subcritical <pitchfork bifurcation>. If $b\ne0$, the branch through zero has $x\sim(1-\mu)/b$ and $M\sim2-\mu$; its stability exchanges with zero at $\mu=1$, a <transcritical bifurcation>. The two nonzero roots meet at
$$
\mu_{\rm sn}=1+b^2/(4a),\qquad x_{\rm sn}=-b/(2a),
$$
where $M=1$. The graph of $\mu(x)$ has nonzero second derivative $-2a$, proving a <saddle-node bifurcation>. At zero and $\mu=-1$, the multiplier is $-1$, indicating a <period-doubling bifurcation>; generically its nondegeneracy coefficient is $a+b^2$. If $a+b^2=0$, necessarily $b\ne0$ here, composition at $\mu=-1$ instead gives $F^2(x)-x=4b^4x^5+O(x^6)$. For $\delta=\mu+1>0$ small, the leading return equation is $-2\delta x+4b^4x^5=0$, producing a two-cycle with leading amplitudes $x\sim\pm[\delta/(2b^4)]^{1/4}$. Its return multiplier is $1+8\delta+o(\delta)>1$. Thus this special case still has a flip bifurcation, but it is degenerate and subcritical rather than the generic cubic normal form.
For $a,b>0$, the fold lies above $\mu=1$. Between it and the transcritical point, the branch $-b/(2a)<x<0$ is stable provided its multiplier remains above $-1$, while the other branch is unstable. For $a,b<0$, the fold lies below $\mu=1$: immediately above it the branch $x<-b/(2a)$ is stable and the branch between $-b/(2a)$ and zero is unstable; the branch crossing zero becomes stable for $\mu>1$. All these statements are subject to the displayed exact multiplier criterion if the diagram is extended. The sketch uses representative parameter values and stops before additional flips, as stipulated.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-4-fixed-point-bifurcations.png]
{title=Fixed-point branches and stability for three representative cubic maps, with folds, transcritical or pitchfork crossings, and the origin flip}
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