If `f(x)=cos[pi^2]x +cos[-pi^2]x ,` where `[x]` stands for the greatest integer function, then

Options

A. `f((pi)/(2))=-1`
B. `f(pi)=1`
C. `f(-pi)=-1`
D. `f((pi)/(4))=2`

Answer

A

Explanation

We have ,
`f(x) =cos[pi^(2)]x+cos[-pi^(2)]x`
`implies f(x)=cos9x + cos (-10x)`
`implies f(x)=cos9x+ cos 10 x`
`:. f((pi)/(2)) =cos""(9x)/(2)+ cos 5 pi =0+(-1)^(5)=-1`
`f(pi) = cos 9 pi + cos 10 pi=(-1)^(9) +(-1)^(10)=-1+1=0`
`f(-pi)= cos (-9pi)+ cos (-10pi) = cos 9 pi + cos 10 pi=0`
` f((pi)/(4))= cos (9pi)/(4)+ cos (10 pi)/(4)=(1)/(sqrt(2))+0=(1)/(sqrt(2))`
Hence , option (a) is correct.