Find the Correct Option
If
\[ f(x)=\sin[\pi^2]x+\sin[-\pi^2]x \]
where \([x]\) denotes the greatest integer less than or equal to \(x\), then
(a) \(f\left(\frac{\pi}{2}\right)=1\)
(b) \(f(\pi)=2\)
(c) \(f\left(\frac{\pi}{4}\right)=-1\)
(d) none of these
Since
\[ \pi^2\approx9.86 \]
\[ [\pi^2]=9 \]
Also,
\[ [-\pi^2]=-10 \]
Therefore,
\[ f(x)=\sin 9x+\sin(-10x) \]
\[ =\sin9x-\sin10x \]
Check option (c):
\[ f\left(\frac{\pi}{4}\right) = \sin\frac{9\pi}{4} – \sin\frac{10\pi}{4} \]
\[ = \sin\frac{\pi}{4} – \sin\frac{5\pi}{2} \]
\[ = \frac1{\sqrt2}-1 \ne -1 \]
Check option (a):
\[ f\left(\frac{\pi}{2}\right) = \sin\frac{9\pi}{2} – \sin5\pi \]
\[ =1-0=1 \]
Hence,
\[ \boxed{\text{Correct Answer: (a)}} \]