c
\(\begin{array}{l}
We\,know\,that,\,v = \frac{{dx}}{{dt}} \Rightarrow dx = v\,dt\\
Integrating\,\int\limits_0^x {dx = \int\limits_0^t {v\,dt} } \\
or\,\,\,\,\,x = \int\limits_0^t {\left( {{v_0} + gt + f{t^2}} \right)dt = \left[ {{v_0}t + \frac{{g{t^2}}}{2} + \frac{{f{t^3}}}{3}} \right]_0^3} \\
or,\,\,\,\,x = {v_0}t + \frac{{g{t^2}}}{2} + \frac{{f{t^3}}}{3}\\
At\,t = 1,\,x = {v_0} + \frac{g}{2} + \frac{f}{3}.
\end{array}\)