If `f(x)=cos{(pi)/(2)[x]-x^(3)},1ltxlt2`, and [x] denotes the greatest integer less than or equal to x, then the value of
`f'(root(3)((pi)/(2)))`, is

Options

A. 0
B. `3((pi)/(2))^(2//3)`
C. `-3((pi)/(2))^(2//3)`
D. none of these

Answer

A

Explanation

In the neighbourhood of `x=root(3)((pi)/(2))` we have `[x]=1`
`:." "f(x)=cos((pi)/(2)-x^(3))`
`implies" "f(x)=sinx^(3)` for all x in the neighbourhood of `root(3)((pi)/(2))`
`implies" "f'(x)=3x^(2)cosx^(3)`
`implies" "f'(root(3)((pi)/(2)))=3((pi)/(2))^(2//3)"cos"(pi)/(4)=0`