Problem

(i) $f(x) = x^3 + 4x - 6$

(a) Show that the equation $f(x) = 0$ has a root $\alpha$ in the interval $[1, 1.5]$

(b) Taking $1.5$ as a first approximation, apply the Newton Raphson process twice to $f(x)$ to obtain an approximate value of $\alpha$. Give your answer to $3$ decimal places. Show your working clearly.

(ii) $g(x) = 4x^2 + x - \tan x$

where $x$ is measured in radians.

The equation $g(x) = 0$ has a single root $\beta$ in the interval $[1.4, 1.5]$

Use linear interpolation on the values at the end points of this interval to obtain an approximation to $\beta$. Give your answer to $3$ decimal places.