ClearAll;
Array[c,5,0]
f[t_]:= c[0] + Apply[Plus,Table[c[i] t^i,{i,5}]];
df[t_]:= Apply[Plus,Table[i c[i] t^(i-1),{i,5}]];
df2[t_]:= Apply[Plus,Table[i(i-1)c[i] t^(i-2),{i,5}]];
eqn1:= f[0] == -1;
eqn2:= f[3] == 5;
eqn3:= f[10] == 4;
eqn4:= df[3] == 0;
eqn5:= df[8] == 0;
eqn6:= df2[5] == 0;
system = {eqn1,eqn2,eqn3,eqn4,eqn5,eqn6}
solution = Solve[system,Table[c[i],{i,0,5}]]
f[t]/.solution
Plot[f[t]/.solution,{t,0,10}, AxesLabel->{"t","y"},
	 PlotLabel->"y = f(t)", AxesOrigin->{-0.5,-1.25},
	 PlotRange->{{0,10},{-1,5}}]