I am solving a problem in this book about Error bound of Lagrange Interpolation.
we have $f(x)=\frac{1}{x}$ is approximated by a second degree polynomial in $[2,\: 4]$ as shown in the screenshot below.
My Query is $3!$ is already accounted in Step $2$
but at the end the author has again taken $3!$ in to account.
Is it a typo?
You are right, it is a typo.
$$\left|\frac{f'''(\xi)}{3!}\right|=\left|\xi(x)\right|^{-4}\le \frac1{16}$$
The upperbound is $$\color{blue}{\left|\frac{f'''(\xi)}{3!}\right|}|(x-x_0)(x-x_1)(x-x_2)| \le \color{blue}{\frac1{16}}\left|-\frac{9}{16}\right|=\frac{9}{256}.$$