To find the equation of the curve, we need to integrate the derivative \(f'(x) = 6x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\).

Integrating term by term:

\(\int 6x^{-\frac{1}{2}} \, dx = 6 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 12x^{\frac{1}{2}}\)

\(\int -4x^{-\frac{3}{2}} \, dx = -4 \cdot \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} = 8x^{-\frac{1}{2}}\)

Thus, the integral of \(f'(x)\) is:

\(f(x) = 12x^{\frac{1}{2}} + 8x^{-\frac{1}{2}} + c\)

We use the point (4, 7) to find \(c\):

\(7 = 12(4)^{\frac{1}{2}} + 8(4)^{-\frac{1}{2}} + c\)

\(7 = 12 \cdot 2 + 8 \cdot \frac{1}{2} + c\)

\(7 = 24 + 4 + c\)

\(c = 7 - 28 = -21\)

Therefore, the equation of the curve is:

\(y = 12x^{\frac{1}{2}} + 8x^{-\frac{1}{2}} - 21\)