To find the equation of the curve, we need to integrate \(\frac{dy}{dx} = 3(4x - 7)^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\).

Integrating the first term:

\(\int 3(4x - 7)^{\frac{1}{2}} \, dx = \frac{3}{2 \times 4} (4x - 7)^{\frac{3}{2}}\)

Integrating the second term:

\(\int -4x^{-\frac{1}{2}} \, dx = -8x^{\frac{1}{2}}\)

Thus, the integrated function is:

\(y = \frac{3}{6}(4x - 7)^{\frac{3}{2}} - 8x^{\frac{1}{2}} + c\)

Using the point \((4, \frac{5}{2})\) to find \(c\):

\(\frac{5}{2} = \frac{1}{2}(9)^{\frac{3}{2}} - 8 \times 4^{\frac{1}{2}} + c\)

\(\frac{5}{2} = \frac{1}{2}(27) - 8 \times 2 + c\)

\(\frac{5}{2} = \frac{27}{2} - 16 + c\)

\(\frac{5}{2} = \frac{27}{2} - \frac{32}{2} + c\)

\(\frac{5}{2} = -\frac{5}{2} + c\)

\(c = 5\)

Therefore, the equation of the curve is:

\(y = \frac{3}{6}(4x - 7)^{\frac{3}{2}} - 8x^{\frac{1}{2}} + 5\)