11 The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(f ^ { \prime \prime } ( x ) = ( 4 x + 1 ) ^ { - \frac { 1 } { 2 } }\).
- Find \(\mathrm { f } ^ { \prime } ( x )\).
It is now given that \(f ^ { \prime \prime } ( 0 ) , f ^ { \prime } ( 0 )\) and \(f ( 0 )\) are the first three terms respectively of an arithmetic progression. - Find the value of \(\mathrm { f } ( 0 )\).
- Find \(\mathrm { f } ( x )\), and hence find the minimum value of f .