(i) Let the height of the cuboid be \(y\). The volume of the cuboid is given by \(3x^2y = 288\), so \(y = \frac{288}{3x^2} = \frac{96}{x^2}\).

The surface area \(A\) is given by:

\(A = 2(x \cdot 3x + x \cdot y + 3x \cdot y) = 2(3x^2 + xy + 3xy).\)

Substitute \(y = \frac{96}{x^2}\) into the equation:

\(A = 2(3x^2 + x \cdot \frac{96}{x^2} + 3x \cdot \frac{96}{x^2}) = 2(3x^2 + \frac{96}{x} + \frac{288}{x}).\)

\(A = 6x^2 + \frac{768}{x}.\)

(ii) To find the stationary value, differentiate \(A\) with respect to \(x\):

\(\frac{dA}{dx} = 12x - \frac{768}{x^2}.\)

Set \(\frac{dA}{dx} = 0\):

\(12x = \frac{768}{x^2}.\)

\(12x^3 = 768.\)

\(x^3 = 64.\)

\(x = 4.\)

Substitute \(x = 4\) back into the expression for \(A\):

\(A = 6(4)^2 + \frac{768}{4} = 96 + 192 = 288.\)

To determine the nature, find the second derivative:

\(\frac{d^2A}{dx^2} = 12 + \frac{1536}{x^3}.\)

At \(x = 4\), \(\frac{d^2A}{dx^2} = 12 + \frac{1536}{64} = 36 > 0\), indicating a minimum.