Subtopic: Solutions of linear and non-linear algebraic equations
Question Statement:
The Newton–Raphson iteration formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ fails completely whenever:
Options:
(A) $f'(x_n) = 0$
(B) $f''(x_n) = 0$
(C) The actual root is negative
(D) The initial guess $x_0$ is greater than the root
Answer Key:
A
Step-by-Step Solution & Explanation
When $f'(x_n) = 0$, the tangent line to the curve at $(x_n, f(x_n))$ is horizontal (parallel to the $x$-axis) and therefore never intersects the $x$-axis. In the formula, this causes division by zero, causing the algorithm to fail.