(a) Differentiate \(y = ax^{\frac{3}{2}} - 12x\) with respect to \(x\):

\(\frac{dy}{dx} = \frac{3}{2}ax^{\frac{1}{2}} - 12\).

Using the chain rule, \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\).

Substitute \(x = 9\) and \(\frac{dx}{dt} = 5\):

\(\frac{dy}{dt} = \left( \frac{3}{2}a \times 9^{\frac{1}{2}} - 12 \right) \times 5\).

\(\frac{dy}{dt} = 5 \left( \frac{9}{2}a - 12 \right) = \frac{45}{2}a - 60 = \frac{15}{2}(3a - 8)\).

(b) For a minimum point at \(x = \frac{1}{4}\), set \(\frac{dy}{dx} = 0\):

\(\frac{3}{2}a \left( \frac{1}{4} \right)^{\frac{1}{2}} - 12 = 0\).

\(\frac{3}{2}a \times \frac{1}{2} - 12 = 0\).

\(\frac{3}{4}a = 12\).

\(a = 16\).