To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = \frac{3}{x^4} + 32x^3\).

Integrating \(\frac{3}{x^4}\) gives \(-\frac{1}{x^3}\).

Integrating \(32x^3\) gives \(8x^4\).

Thus, the general solution is \(y = -\frac{1}{x^3} + 8x^4 + c\).

We use the point \(\left( \frac{1}{2}, 4 \right)\) to find \(c\).

Substitute \(x = \frac{1}{2}\) and \(y = 4\) into the equation:

\(4 = -\frac{1}{(\frac{1}{2})^3} + 8(\frac{1}{2})^4 + c\)

\(4 = -8 + \frac{1}{2} + c\)

\(4 = -8 + 0.5 + c\)

\(4 = -7.5 + c\)

\(c = 4 + 7.5 = 11.5\)

Therefore, the equation of the curve is \(y = -\frac{1}{x^3} + 8x^4 + \frac{23}{2}\).