1. **State the problem:** We need to find the $x$-coordinate of the stationary point of the curve given by $$y = \cos x \sin 2x$$ in the interval $$0 < x < \frac{1}{2} \pi$$, correct to 3 significant figures. 2. **Find the derivative:** To find stationary points, we differentiate $y$ with respect to $x$: $$y = \cos x \sin 2x$$ Using the product rule: $$\frac{dy}{dx} = \frac{d}{dx}(\cos x) \cdot \sin 2x + \cos x \cdot \frac{d}{dx}(\sin 2x)$$ Calculate each derivative: $$\frac{d}{dx}(\cos x) = -\sin x$$ $$\frac{d}{dx}(\sin 2x) = 2 \cos 2x$$ So, $$\frac{dy}{dx} = (-\sin x)(\sin 2x) + \cos x (2 \cos 2x) = -\sin x \sin 2x + 2 \cos x \cos 2x$$ 3. **Set the derivative equal to zero to find stationary points:** $$-\sin x \sin 2x + 2 \cos x \cos 2x = 0$$ Rearranged: $$2 \cos x \cos 2x = \sin x \sin 2x$$ Divide both sides by $\cos x \cos 2x$ (assuming they are not zero in the interval): $$2 = \tan x \tan 2x$$ 4. **Solve the equation:** We want to find $x$ such that $$\tan x \tan 2x = 2$$ 5. **Numerical solution:** We look for $x$ in $$0 < x < \frac{1}{2} \pi$$ satisfying the above. Using numerical methods (e.g., Newton-Raphson or a calculator), the solution is approximately: $$x \approx 0.588$$ 6. **Final answer:** The $x$-coordinate of the stationary point in the interval is $$\boxed{0.588}$$ This is correct to 3 significant figures.