Newton-Raphson convergence failure

A question is this type if and only if it asks to explain why Newton-Raphson will not converge or why a particular starting value cannot be used, typically involving analysis of f'(x) = 0 or geometric interpretation.

2 questions

Edexcel F1 2023 January Q4
4. $$f ( x ) = 1 - \frac { 1 } { 8 x ^ { 4 } } + \frac { 2 } { 7 \sqrt { x ^ { 7 } } } \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\), that lies in the interval \([ 0.15,0.25 ]\)
    1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
    2. Explain why 0.25 cannot be used as an initial approximation for \(\alpha\) in the Newton-Raphson process.
    3. Taking 0.15 as a first approximation to \(\alpha\) apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\) Give your answer to 3 decimal places.
  2. Use linear interpolation once on the interval \([ 0.15,0.25 ]\) to find another approximation to \(\alpha\) Give your answer to 3 decimal places.
OCR FP2 2008 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-3_606_890_815_630} The diagram shows the curve with equation \(y = x \mathrm { e } ^ { - x } + 1\). The curve crosses the \(x\)-axis at \(x = \alpha\).
  1. Use differentiation to show that the \(x\)-coordinate of the stationary point is 1 .
    \(\alpha\) is to be found using the Newton-Raphson method, with \(\mathrm { f } ( x ) = x \mathrm { e } ^ { - x } + 1\).
  2. Explain why this method will not converge to \(\alpha\) if an initial approximation \(x _ { 1 }\) is chosen such that \(x _ { 1 } > 1\).
  3. Use this method, with a first approximation \(x _ { 1 } = 0\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Find \(\alpha\), correct to 3 decimal places.