Problem

$f(x) = 1 - \frac{1}{8x} + \frac{2}{4x^2} \quad x > 0$

The equation $f(x) = 0$ has a single root, $\alpha$, that lies in the interval $[0.15, 0.25]$

(a) (i) Determine $f'(x)$

(ii) Explain why $0.25$ cannot be used as an initial approximation for $\alpha$ in the Newton-Raphson process.

(iii) Taking $0.15$ as a first approximation to $\alpha$ apply the Newton-Raphson process once to $f(x)$ to obtain a second approximation to $\alpha$ Give your answer to $3$ decimal places.

(b) Use linear interpolation once on the interval $[0.15, 0.25]$ to find another approximation to $\alpha$ Give your answer to $3$ decimal places.