Solution
= Solution
The <Newton root-finding iteration> replaces the graph by its <tangent line> at $x_0$ and uses that line's zero:
$$
0=f(x_0)+f'(x_0)(x_1-x_0),\qquad
x_1=x_0-\frac{f(x_0)}{f'(x_0)}.
$$
Comparing with the printed reciprocal-ratio form gives
$$
\boxed{\alpha=-1,\qquad\beta=-1}.
$$
This step requires $f'(x_0)\ne0$. If $f(x_0)=0$, the iteration has already reached a <zero of a function> and the printed reciprocal-ratio expression need not be evaluated.