Discussion
The tangent line to \(y = f(x)\) at the point \((x_{0}, y_{0})\), assuming \(f'(x) \neq 0\), intersects the x axis at
Answer: A
The tangent meets the x axis at \(\left(x_{0} - y_{0}/f'(x_{0}),\ 0\right)\).
The tangent at \((x_{0}, y_{0})\) has slope \(f'(x_{0})\):
\[ \begin{aligned} y - y_{0} &= f'(x_{0})(x - x_{0}) \\ \text{at } y = 0:\ x &= x_{0} - \frac{y_{0}}{f'(x_{0})} \end{aligned} \]
The condition \(f'(x) \neq 0\) is what guarantees the tangent is not horizontal and does cut the axis. This is the step used in Newton-Raphson iteration. The option with a plus sign has the sign of the correction wrong, and the two options with \(f'(x_{0})/y_{0}\) invert the ratio.
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