Practice question · Multiple choice
Newton's method usually converges spectacularly fast and sometimes flies off entirely. What single feature of the step explains both behaviours?
Hints
- Write the step: x − f(x)/f'(x). Ask what happens when f'(x) is close to zero.
- The same formula produces both behaviours. Look for the term that can misbehave.
Show the answer
A. The division by f'(x), which throws the guess far if the slope is flat
Why
One term does both jobs: a steep tangent points almost exactly at the root and the correct digits roughly double each step, while a flat one divides by nearly zero and launches the iterate somewhere useless. That is why production solvers keep a bracketing method underneath, speed when it works, a guarantee when it does not.
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