(a) Differentiate \(y = \sqrt{x} \cos x\) using the product rule: \(\frac{dy}{dx} = \frac{1}{2\sqrt{x}} \cos x - \sqrt{x} \sin x\).

Set the derivative to zero for the minimum point: \(\frac{1}{2\sqrt{x}} \cos x = \sqrt{x} \sin x\).

Simplify to \(\cos x = 2x \sin x\), leading to \(\tan x = \frac{1}{2x}\).

(b) Use the iterative formula \(a_{n+1} = \pi + \arctan\left(\frac{1}{2a_n}\right)\) starting with \(a_1 = 3\).

Calculate: \(a_2 = 3.3067\), \(a_3 = 3.2917\), \(a_4 = 3.2923\).

The value converges to 3.29 correct to 2 decimal places.

(c) The volume of the solid is given by \(\pi \int (\sqrt{x} \cos x)^2 \, dx\).

Use integration by parts and trigonometric identities to find the integral: \(\frac{x^2}{4} + \frac{x}{4} \sin 2x - \frac{1}{4} \int \sin 2x \, dx\).

Complete the integration to obtain \(\frac{1}{4} x^2 + \frac{1}{4} x \sin 2x + \frac{1}{8} \cos 2x\).

Substitute the limits \(x = 0\) and \(x = \frac{3}{2}\pi\) to find the volume: \(\frac{1}{16} \pi (\pi^2 - 4)\).