2024 RI P1 Q9

A curve \(C\) has equation \(y=2ax+\frac{b}{x}\) where \(a\) and \(b\) are non-zero real constants and \(x\ne 0\).

1. Using differentiation, determine whether \(C\) has any stationary points if \(ab < 0\).[2]

It is now given that \(b=\frac{1}{2}a\) where \(a>0\).

2. Sketch \(C\), stating the equations of any asymptotes, and the coordinates of any stationary points and points where \(C\) crosses the axes (if any).[3]
3. State the range of values of \(k\), in terms of \(a\), for which the equation \(2ax+\frac{b}{x}=kx\) has no real roots.[1]
4. The region bounded by \(C\), the axes, the lines \(x=\frac{1}{2}\) and \(y=4a\) is rotated about the \(y\)-axis through \(2\pi \) radians. Show that the volume generated is given by \(\frac{1}{2}a\pi +\frac{\pi }{8{{a}^{2}}}\displaystyle\displaystyle \displaystyle\int_{2a}^{4a}{{{y}^{2}}}-2{{a}^{2}}-y\sqrt{{{y}^{2}}-4{{a}^{2}}}\ \mathrm{d}y\).
   Hence, find, in terms of \(a\) and \(\pi \), the exact volume generated.[6]