Question:
If f(x) = sin [π2] x + sin [−π]2 x, where [x] denotes the greatest integer less than or equal to x, then
(a) f(π/2) = 1
(b) f(π) = 2
(c) f(π/4) = −1
(d) None of these
Solution:
(a) f(π/2) = 1
$f(x)=\sin \left[\pi^{2}\right] x+\sin \left[-\pi^{2}\right] x$
$\Rightarrow f(x)=\sin [9.8] x+\sin [-9.8] x$
$\Rightarrow f(x)=\sin 9 x-\sin 10 x$
$f\left(\frac{\pi}{2}\right)=\sin 9 \times \frac{\pi}{2}-\sin 10 \times \frac{\pi}{2}$
$\Rightarrow f\left(\frac{\pi}{2}\right)=1-0=1$