(i) To find the stationary point, we first find the derivative of the function:

\(\frac{dy}{dx} = 6e^x - 3e^{3x}\).

Set the derivative equal to zero to find the stationary point:

\(6e^x - 3e^{3x} = 0\).

Factor out \(3e^x\):

\(3e^x(2 - e^{2x}) = 0\).

Since \(3e^x \neq 0\), we have:

\(2 - e^{2x} = 0\).

Thus, \(e^{2x} = 2\).

Taking the natural logarithm of both sides gives:

\(2x = \ln 2\).

Therefore, \(x = \frac{1}{2} \ln 2\).

(ii) To determine the nature of the stationary point, consider the second derivative:

\(\frac{d^2y}{dx^2} = 6e^x - 9e^{3x}\).

Evaluate the second derivative at \(x = \frac{1}{2} \ln 2\):

\(\frac{d^2y}{dx^2} = 6e^{\frac{1}{2} \ln 2} - 9e^{3(\frac{1}{2} \ln 2)}\).

\(= 6 \sqrt{2} - 9 \cdot 2^{\frac{3}{2}}\).

\(= 6 \sqrt{2} - 18 \sqrt{2}\).

\(= -12 \sqrt{2}\).

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