Wait, but g'''(0)=14
integrate that, g''(x)=14x+C, C=0 since g''(0)=0 through substitution
integrate, g'(x)=7x2+C, again C=0 through substitution
integrate, g(x)=7x3/3+C, C=0 through substitution
then plug that g(x) into the piecewise function to get f(x)=7x/3
Seems to work, doesn't it?