# unique fixed point withnumerical calculus

Let $g(x) =\frac{2}{3} (x+\frac{1}{x^2})$, show that $g$ has a unique fixed point $r\in [1,2]$. what is it?

If $x_0∈[1,2]$, show that the functional iterates converge to $r$, $x_n→r$, and give the rate of convergence?

help me understand this i am pretty sure it's a contractive mapping theorem question but i am having a hard time understanding the procedure.

• Tell us how far you got in computing $\frac{g(y)-g(x)}{y-x}$. – LutzL Oct 29 '17 at 5:32

Contraction mapping theorem? Don't use bombs to kill ants. By definition of a fixed point, we have to solve $$x = g(x) = \frac23 \left(x + \frac{1}{x^2} \right),$$ so we have to solve $$x^3 = 2.$$ This equation a unique solution in $[1,2]$, namely $\sqrt[3]{2}$.