(i) To find the coordinates of \(A\), set \(y = 0\) in the equation \(y = 4\sqrt{x} - x\):

\(4\sqrt{x} - x = 0\)

\(4\sqrt{x} = x\)

\(4 = \sqrt{x}\)

\(x = 16\)

Thus, \(A(16, 0)\).

To find the coordinates of \(M\), find the derivative \(\frac{dy}{dx} = 2x^{-\frac{1}{2}} - 1\) and set it to zero:

\(2x^{-\frac{1}{2}} - 1 = 0\)

\(2\sqrt{x} = 1\)

\(\sqrt{x} = \frac{1}{2}\)

\(x = 4\)

Substitute \(x = 4\) back into the original equation:

\(y = 4\sqrt{4} - 4 = 4\)

Thus, \(M(4, 4)\).

(ii) The volume \(V\) of the solid of revolution is given by:

\(V = \pi \int y^2 \, dx\)

\(V = \pi \int (4\sqrt{x} - x)^2 \, dx\)

\(V = \pi \int (16x + x^2 - 8x^{\frac{3}{2}}) \, dx\)

Integrate each term:

\(\pi \left[ 8x^2 + \frac{x^3}{3} - 8\frac{x^{\frac{5}{2}}}{\frac{5}{2}} \right]\)

Evaluate from 0 to 16:

\(V = \pi \left[ 8(16)^2 + \frac{(16)^3}{3} - 8\frac{(16)^{\frac{5}{2}}}{\frac{5}{2}} \right]\)

\(V = 136.5\pi\) (or \(137\pi\))