Problem $3.($Fixed point method$)$

We consider the solution of the equation

$cos(e^{-x})=2\sqrt[\phantom{2}]{x}$

a$)$ Show that the following fixed point method

$x=g(x)$ with $g(x)=\frac{co{s}^2(e^{-x})}{4}$

has a unique solution hat$(x)0.$

b$)$ If you run the code below, you will obtain a value $x$ that is a numerical approximation

of hat$(x).$ Provide an upper bound for the error $e$:$=|hat(x)-x|.$

import numpy as $np$

def $g(x)$ :

return np$.cos(np*exp(-x){)}^{****}\frac{2}{4}$

$x=0$

$x_{?}$old $=1$

while np$.$abs old $\{$:$-x)>1e-6$ :

$x_{?}$old $=x$

$x=g(x)$

print $($x$)$