Solution:

The function must increase by a factor of 27 when $x$ increases by $c$. Since $27 = 3^3$, the base of the exponential can be expressed as $3^3$, and the coefficient of $x$ in the exponent must include $\frac{1}{c}$ as a factor to satisfy the condition. Therefore, the correct form of the function is:

$$f(x) = 48(3^3)^{\frac{1}{6}x}$$

Answer: B) $f(x) = 48(3^3)^{\frac{1}{6}x}$