To find the stationary points, we need to find where the derivative \(\frac{dy}{dx}\) is zero.

Using the product rule, the derivative of \(y = \cos x \sin 2x\) is:

\(\frac{dy}{dx} = -\sin x \sin 2x + 2\cos x \cos 2x\).

Using the double angle formula \(\cos 2x = 1 - 2\sin^2 x\), express the derivative in terms of \(\sin x\) and \(\cos x\):

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

Equate the derivative to zero:

\(-\sin x \sin 2x + 2\cos x \cos 2x = 0\).

Simplify to obtain an equation in one trigonometric function:

\(3\sin 2x = 1\) or \(3\cos 2x = 2\) or \(2\tan 2x = 1\).

Solving \(2\tan 2x = 1\) gives:

\(\tan 2x = \frac{1}{2}\).

Solving for \(x\) in the interval \(0 < x < \frac{1}{2} \pi\), we find:

\(x = 0.615\).