(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.
