(i) To find the stationary points, we differentiate the function \(y = 10e^{-\frac{1}{2}x} \sin 4x\) using the product rule:

\(\frac{dy}{dx} = -5e^{-\frac{1}{2}x} \sin 4x + 40e^{-\frac{1}{2}x} \cos 4x\).

Setting \(\frac{dy}{dx} = 0\), we solve:

\(\tan 4x = 8\).

This gives \(x = 0.362\) or \(x = 1.147\) (in radians).

(ii) The \(x\)-coordinates of \(T_n\) increase by \(\frac{\pi}{4}\). We solve the inequality:

\(x_1 + (n-1)\frac{\pi}{4} > 25\).

Substituting \(x_1 = 0.362\), we get:

\(n > \frac{4}{\pi}(25 - 0.362) + 1\).

Calculating gives \(n = 33\).