(i) The total surface area of the block is given by the formula for the surface area of a rectangular prism: \(4xh + 2x^2 = 96\).

Solving for \(h\), we have:

\(4xh + 2x^2 = 96\)

\(4xh = 96 - 2x^2\)

\(h = \frac{96 - 2x^2}{4x} = \frac{24}{x} - \frac{x}{2}\)

The volume \(V\) of the block is \(x^2h\):

\(V = x^2 \left( \frac{24}{x} - \frac{x}{2} \right)\)

\(V = 24x - \frac{1}{2}x^3\)

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

\(\frac{dV}{dx} = 24 - \frac{3}{2}x^2\)

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

\(24 - \frac{3}{2}x^2 = 0\)

\(\frac{3}{2}x^2 = 24\)

\(x^2 = 16\)

\(x = 4\)

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

\(V = 24(4) - \frac{1}{2}(4)^3 = 96 - 32 = 64\)

(iii) To determine if this is a maximum or minimum, find the second derivative:

\(\frac{d^2V}{dx^2} = -3x\)

Substitute \(x = 4\):

\(\frac{d^2V}{dx^2} = -3(4) = -12\)

Since \(\frac{d^2V}{dx^2} < 0\), the stationary point is a maximum.