To find the stationary point, we need to differentiate the function \(y = \cos^3 x \sqrt{\sin x}\) and set the derivative equal to zero.

Using the product rule and chain rule, the derivative is:

\(\frac{dy}{dx} = -3\cos^2 x \sin x \sqrt{\sin x} + \frac{\cos^3 x \cos x}{2\sqrt{\sin x}}\).

Simplifying, we get:

\(-3\cos^2 x \sin^2 x + \frac{1}{2}\cos^4 x = 0\).

Rearranging gives:

\(7\cos^3 x = 6\).

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

\(x = 0.388\).