Let $f(x) = \dfrac{\sin \left([x ]\right)}{a} + \dfrac{\cos\left([x ] \right)}{a}$, where $a>0$ and $[ \cdot ]$ denotes the fractional part of $x$. Then the set of values of $a$ for which $f$ can attain its maximum value is
a) $(0,4/π)$
b)$(4/π,∞)$
c)$(0,∞)$
d) none of these
One can write $f(x)=\dfrac{g(x)}{a}$ with $g(x):=sin([x])+cos([x]).$
The answer is visibly independent of $a$ because the maximum of $\dfrac{g(x)}{a}$ occurs for the same values of $x$ as the maximum of $g(x)$.
As one can check that $g(x)=\sqrt{2}\cos([x]-\frac{\pi}{4})$, it is a period-1 function. It is enough to study it on [0,1]. On this interval $g(x)$ coincides with $h(x):=\sqrt{2}\cos(x-\frac{\pi}{4})$ as shown on the graphics below. Therefore, the maximum value occurs for $x=\frac{\pi}{4}$.
The set of positive solutions is then $S=\{ \frac{\pi}{4}+k \ | \ k \in \mathbb{N} \}$
It does not coincide with any of the first three solutions... thus it's the fourth one.