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\includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-18_611_531_262_808} The diagram shows the curve \(y = \mathrm { f } ( x )\) defined for \(x > 0\). The curve has a minimum point at \(A\) and crosses the \(x\)-axis at \(B\) and \(C\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - \frac { 2 } { x ^ { 3 } }\) and that the curve passes through the point \(\left( 4 , \frac { 189 } { 16 } \right)\).

1. Find the \(x\)-coordinate of \(A\).
2. Find \(\mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of \(B\) and \(C\).
4. Find, showing all necessary working, the area of the shaded region.
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