(i) To find \(\frac{dy}{dx}\), differentiate \(y = \frac{8}{x} + 2x\):

\(\frac{dy}{dx} = -8x^{-2} + 2\).

To find \(\frac{d^2y}{dx^2}\), differentiate \(\frac{dy}{dx}\):

\(\frac{d^2y}{dx^2} = 16x^{-3}\).

To find \(\int y^2 \, dx\), integrate \(y^2 = \left(\frac{8}{x} + 2x\right)^2\):

\(\int y^2 \, dx = -64x^{-1} + 32x + \frac{4x^3}{3} + C\).

(ii) Set \(\frac{dy}{dx} = 0\) to find stationary points:

\(-8x^{-2} + 2 = 0 \Rightarrow x = \pm 2\).

For \(x > 0\), \(M(2, 8)\).

For \(x < 0\), the other turning point is \((-2, -8)\).

Check the nature using \(\frac{d^2y}{dx^2}\):

At \(x = -2\), \(\frac{d^2y}{dx^2} < 0\), so it is a maximum.

(iii) The volume of revolution is given by:

\(V = \pi \int_{1}^{2} y^2 \, dx\).

Using the integral from part (i),

\(V = \pi \left[ -64x^{-1} + 32x + \frac{4x^3}{3} \right]_{1}^{2}\).

Calculate the definite integral:

\(V = \frac{220\pi}{3}\).