2.
$$f ( x ) = 3 x ^ { \frac { 3 } { 2 } } - 25 x ^ { - \frac { 1 } { 2 } } - 125 , \quad x > 0$$
- Find \(f ^ { \prime } ( x )\).
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [12, 13].
- Using \(x _ { 0 } = 12.5\) 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 decimal places.
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