The diagram shows the curve \(y = f(x)\) 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 \(\frac{dy}{dx} = 2x - \frac{2}{x^3}\) and that the curve passes through the point \(\left(4, \frac{189}{16}\right)\).

(i) Find the \(x\)-coordinate of \(A\).

(ii) Find \(f(x)\).

(iii) Find the \(x\)-coordinates of \(B\) and \(C\).

(iv) Find, showing all necessary working, the area of the shaded region.