The curve \(C\) has equation
\(y = \frac{ax^2 + x}{x + b},\)
where \(a\) and \(b\) are positive constants such that \(a \neq \frac{1}{b}\).
Given that \(C\) has no stationary points, use differentiation to find the relationship between \(a\) and \(b\).[4]
It is now given that \(a = 2\) and \(b = 1\).
Sketch \(C\), stating the equations of any asymptotes, coordinates of any turning points and of the points where \(C\) crosses the axes.[4]
Hence find the volume of the solid formed when the finite region bounded by \(C\) and the \(x\)-axis is rotated completely about the \(x\)-axis.[2]