(i) Differentiate \(y = x \ln(8 - x)\) using the product rule: \(\frac{dy}{dx} = \ln(8 - x) - \frac{x}{8 - x}\).

Set \(\frac{dy}{dx} = 1\) and solve: \(\ln(8 - x) - \frac{x}{8 - x} = 1\).

Rearrange to find \(x = 8 - \frac{8}{\ln(8 - x)}\).

(ii) Calculate \(8 - \frac{8}{\ln(8 - 2.9)} = 3.09\) and \(8 - \frac{8}{\ln(8 - 3.1)} = 2.97\).

Since \(3.09 > 2.9\) and \(2.97 < 3.1\), \(a\) lies between 2.9 and 3.1.

(iii) Use the iterative formula \(x_{n+1} = 8 - \frac{8}{\ln(8 - x_n)}\).

Starting with \(x_0 = 3\), iterate: \(x_1 = 3.0293\), \(x_2 = 3.0111\), \(x_3 = 3.0223\), \(x_4 = 3.0154\), \(x_5 = 3.0198\).

Consecutive values \(3.0223\) and \(3.0154\) round to 3.02, confirming \(a = 3.02\).