Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
The curve \(C\) is translated by \(\left[ \begin{array} { l } 0 k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
(1 mark)