1.
$$\mathrm { f } ( x ) = 6 \sqrt { x } - x ^ { 2 } - \frac { 1 } { 2 x } , \quad x > 0$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 3,4 ]\).
- Taking 3 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.
[0pt] - Use linear interpolation once on the interval [3,4] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.