[Math] Using fixed point iteration find the positive root of $f(x)=e^{-x}-x^2$

Question

Consider $f(x)=e^{-x}-x^2$.

I'm suppose to find the positive root using fixed point iteration.

after drawing the graph, it's safe to set the interval from [0.25,1].
(I actually want to set it from [0.5,1] but i'm unsure as i can't zoom into the graph).

The problem i'm having is deriving the function $g$.

I got $g(x)=-2\ln(x)$.

I can't seem to prove the uniqueness of this root as $g'(x)=\frac{-2}{x}$ but i can bound $|g'(x)|<K$ such that K<1 under this interval. Any help is appreciated.

I've tried to think of other $g(x)$ but none of them seem convergent to me

Answer 1

I think $g(x)=e^{-x/2}$ works.