To find \(f(x)\), we need to integrate \(f'(x) = 2x^{-\frac{1}{3}} - x^{\frac{1}{3}}\).

Integrating term by term:

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

\(\int x^{\frac{1}{3}} \, dx = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4}x^{\frac{4}{3}}\)

Thus, \(f(x) = 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + c\).

Using the condition \(f(8) = 5\):

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

\(5 = 12 - 12 + c\)

\(c = 5\)

Therefore, \(f(x) = 3x^{\frac{2}{3}} - \frac{3}{4}x^{\frac{4}{3}} + 5\).