(i) To find the \(x\)-coordinate of \(M\), we need to find the maximum point of the function \(y = e^{-\frac{1}{2}x^2} \sqrt{(1 + 2x^2)}\). Differentiate \(y\) with respect to \(x\) using the product and chain rules. Set the derivative to zero and solve for \(x\). The solution is \(x = \frac{1}{\sqrt{2}}\).

(ii) The iterative formula \(x_{n+1} = \sqrt{(\ln(4 + 8x_n^2))}\) converges to \(\alpha\). Therefore, \(\alpha = \sqrt{(\ln(4 + 8\alpha^2))}\). Rearrange to find \(e^{\alpha^2} = 4 + 8\alpha^2\). Substitute \(y = 0.5\) into the original curve equation and show that \(\alpha\) satisfies this equation.

(iii) Use the iterative formula starting with \(x_1 = 2\) and calculate successive values until the result stabilizes to 2 decimal places. The iterations yield \(\alpha = 1.86\).