3.
$$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } - 5 , \quad x > 0$$
- Find \(\mathrm { f } ^ { \prime } ( x )\).
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4.5, 5.5].
- Using \(x _ { 0 } = 5\) as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 significant figures.