Problem

5. $f(x) = x^4 - 12x^2 + 7 \quad x > 0$

(a) Show that the equation $f(x) = 0$ has a root, $\alpha$, in the interval $[2, 3]$.

(b) Taking $2.5$ as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $f(x)$ to find a second approximation to $\alpha$, giving your answer to 2 decimal places.

(c) Show that your answer to (b) gives $\alpha$ correct to 2 decimal places.