The function $f$ is differentiable for all real numbers. The point $\left(3, \dfrac{1}{4}\right)$ is on the graph of $y = f(x)$, and the slope at each point $(x, y)$ on the graph is given by $\dfrac{dy}{dx} = y^{2}(6 - 2x)$.
(a) Find $\dfrac{d^{2}y}{dx^{2}}$ and evaluate it at the point $\left(3, \dfrac{1}{4}\right)$.
(b) Find $y = f(x)$ by solving the differential equation $\dfrac{dy}{dx} = y^{2}(6 - 2x)$ with the initial condition $f(3) = \dfrac{1}{4}$.