EDIT:
Now I see that my attempt isn't get $f(x)=0$,
then I think my solution is correct , I will be grateful for some help.

$f(x)=x^5-x^3+3$.

Find an interval such that $\forall x\_0$ in this interval , the newton raphson method is convergent.

I know this thoerem:

$f\in C^2(\mathbb{R}), \forall x\in \mathbb{R},f'(x)>0,f''(x)>0 \implies \forall x\_0$ the the newton raphson method is convergent.

In my problem:

The graph of $f(x)=x^5-x^3+3$ is :
[![enter image description here](https://i.stack.imgur.com/kUFwC.png)](https://i.stack.imgur.com/kUFwC.png)

The interval $(\sqrt{\frac{3}{5}},\infty)$ exists $ f'(x)>0,f''(x)>0 , \forall x\in (\sqrt{\frac{3}{5}},\infty)$.

Is it correct to say that the newton raphson method is convergent $\forall x\in (\sqrt{\frac{3}{5}},\infty)$.

In case I am wrong how to approach this problem ?

Thanks !