This is a problem from the book "Numerical Methods for Unconstrained Optimization and Nonlinear Equations" by J. E. Dennis, Jr., Robert B. Schnabel.
For each of the functions $f_1(x)=x, f_2(x)=x^{2}+x, f_3(x)=e^{x}-1,$ answer the following questions:
(a) What is $f^{\prime}(x)$ at the root $x_{*}=0$?
(b) What is a Lipschitz constant for $f^{\prime}(x)$ in the interval $[-a, a] ;$ i.e., what is a bound on $\left|\left(f^{\prime}(x)-f^{\prime}(0)\right) / x\right|$ in this interval?
(c) What region of convergence of Newton's method on $f(x)$?
(d) In what interval $[b, c], b<0<c,$ is Newton's method on $f(x)$ actually convergent to $x_{*}=0 ?$
This is my progress so far:
(a) It's easy to have $f^{\prime}_1(x)=1, f^{\prime}_2(x)=2x+1, f^{\prime}_3(x)=e^{x}$, therefore $f^{\prime}(x_*)=1$ for all these functions.
(b) I'm not very sure from here. For the first function, $\left| \frac{f^{\prime}(x)-1}{x} \right| = \left| \frac{1-1}{x} \right| \leq 0, \forall x \in [-a, a]$. For the second function, $\left| \frac{f^{\prime}(x)-1}{x} \right| = \left| \frac{2x}{x} \right| \leq 2, \forall x \in [-a, a]$. For the third function, $\left| \frac{f^{\prime}(x)-1}{x} \right| = \left| \frac{e^x-1}{x} \right| \leq \frac{e^a-1}{a}, \forall x \in [-a, a]$. Therefore, the Lipchitz constant is $0$, $2$ and $\frac{e^a-1}{a}$ respectively.
(c) and (d) I am not sure how to start with those questions.