`I ff(x)=2tan^(- 1)x+sin^(- 1)((2x)/(1+x^2)), x > 1`.T h e n , f(5) is equal to

Options

A. `f'(2)=f'(3)`
B. `f'(2)=0`
C. `f'((1)/(2))=(16)/(5)`
D. all the above

Answer

D

Explanation

We know that
`sin^(-1)((2x)/(1+x^(2)))={{:(2tan^(-1)x" ,"," if "-1lexle1),(pi-2tan^(-1)x" ,"," if "xgt1),(-pi-2tan^(-1)x","," if "xlt-1):}`
`:." "f(x)={{:(4tan^(-1)x" ,"," if "-1lexle1),(" "pi" ,"," if "xgt1),(-pi" ,"," if "xlt-1):}`
`implies" "f'(x)={{:((4)/(1+x^(2))","," if "-1ltxlt1),(0" ,"," if "|x|gt1):}`
`implies" "f'(2)=f'(3)=0" and "f'((1)/(2))=(16)/(5)`