Are the fixed points of $$u_{n+1}=\frac15\left(4u_n+\frac3{u_n}-2\right)$$ where $n\in\Bbb N$, stable?
My friend and I are disagreeing but I haven't seen his work. I have obtained that the two fixed points are stable and my friend got one stable and one unstable. Who is right, if anyone?
Let $f(x) = (4x + 3/x - 2)/5$. The fixed points are the solutions of $f(x) = x$, namely $-3$ and $1$, at which $f'$ has the values $11/15$ and $1/5$ respectively. Since these have absolute value $< 1$, they are stable.