The period of the function $\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}+3-[3\mathrm{x}+3]+\sin \frac{\mathrm{\pi x}}{2}$, where [x] denotes the greatest integer ≤ x, is

3x + 3 – [3x + 3] has the period 1 and sin $\frac{\mathrm{\pi x}}{2}$ has the period $\frac{2\mathrm{\pi }}{\mathrm{\pi }/2}$ i.e., 4. Therefore, the period of f (x) is L.C.M. (1, 4) = 4.