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I've solved the first part. I think I have something for the second part, but I'm unsure.

A) You are given the fixed point problem $x=Ax^2$ where $A>0$ is a constant. Compute positive fixed point of the problem.

The fixed point is $1/A$, because $1/A=A\, (1/A)^2$.

B) For what values of $A>0$ will a fixed point iteration $p_n=A(p_{n-1})^2 $ converge to the positive fixed point?

I think this is from $0$ to $A=1/2$, $(0,1/2)$ but am unsure. This is because the convergence theorem. If we set $g(x)=Ax^2$ and compute derivative $2Ax$, then $2Ax\leq K<1$ according to the convergence theorem, the derivative has to be less than some constant $K$ which is less than $1$ on the interval of convergence. Therefore $Ax<1/2$ $\implies$ $A<1/2$ for the sequence to converge.

Is this right?

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The derivative of $A\,x^2$ evaluated at $x=1/A$ is $2$ for all $A$. The fixed point iteration will never converge to $1/A$ except if $x_0=1/A$. If $0\le x_0<1/A$ it will converge to the aerator $0$. If $x>1/A$ it will converge to $\infty$.

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