Find the Value of \( f(\pi) \)
Question:
If
\[ f(x)=\cos[\pi^2]x+\cos[-\pi^2]x \]
where \([x]\) denotes the greatest integer less than or equal to \(x\), then write the value of
\[ f(\pi) \]
Solution:
Since
\[ \pi^2\approx9.86 \]
\[ [\pi^2]=9 \]
and
\[ [-\pi^2]=-10 \]
Therefore,
\[ f(x)=\cos9x+\cos(-10x) \]
\[ =\cos9x+\cos10x \]
Now,
\[ f(\pi)=\cos9\pi+\cos10\pi \]
\[ =-1+1 \]
\[ =0 \]
Therefore,
\[ \boxed{f(\pi)=0} \]