c
\(\begin{array}{l}
Give\,:\,At\,time\,t = 0,\,velocity,\,v = 0\\
Accelerations\,f\, = \,{f_0}\left( {1 - \frac{t}{T}} \right)\\
At\,f = 0,\,\,\,\,0 = {f_0}\left( {1 - \frac{t}{T}} \right)\\
\sin ce\,{f_0}\,is\,a\,constant,\\
\therefore \,1 - \frac{t}{T} = 0\,\,\,or\,\,t = T.
\end{array}\)

\(\begin{array}{l}
Also,\,acceleation\,f = \frac{{dv}}{{dt}}\\
\therefore \,\int\limits_0^{{v_x}} {dv}  = \int\limits_{t = 0}^{t = T} {fdt = \int\limits_0^T {{f_0}} } \left( {1 - \frac{t}{T}} \right)dt\\
\therefore \,\,{v_x} = \left[ {{f_0}t - \frac{{{f_0}{t^2}}}{{2T}}} \right]_0^T = {f_0}T - \frac{{{f_0}{T^2}}}{{2T}} = \frac{1}{2}{f_0}T
\end{array}\)