(Alternative solution) By Hint 2, we know that and are -periodic.
This implies that for any constant and , both functions and are also -periodic;
Indeed, putting , we have .
Similar argument holds for .
Therefore, (i), (ii), (iii) are -periodic, while (iv), (v), (vi) are -periodic.
However, since at (6 am) and at (6 pm), the function values of are different, i.e., ,
is not -periodic---(i), (ii), (iii) are out.
Then, using , given in (v) and (vi) satisfies , which is different from the given information ---(v), (vi) are out.
Answer