If `f(x-y), f(x) f(y) and f(x+y)` are in A.P. for all `x , y ,and f(0)!=0,` then (a) `f(4)=f(-4)` (b) `f(2)+f(-2)=0` (c) `f^(prime)(4)+f^(prime)(-4)=0` (d) `f^(prime)(2)=f^(prime)(-2)`

Options

A. `f'(2)=f'(2)`
B. `f'(-3)=-f'(3)`
C. `f'(-2)+f'(2)=0`
D. none of these

Answer

A

Explanation

It is given that for all `x, y in R`
`f(x-y), f(x) f(y) and f(x+y)"are in A.P"`
`therefore 2f(x) f(y)=f(x-y)+f(x+y)"for all "x,y in R`
`Rightarrow 2f(0) f(x)=f(-x)+f(x)"for all "x in R`[Replacing x by 0 and y by x]
`Rightarrow f(-x)+f(x)="for all "x in R`
`Rightarrow f(-x)=-f(x)="for all "x in R`
`Rightarrow -f'(-x)=-f'(x)="for all "x in R`
`Rightarrow f'(-x)=f'(x)="for all "x in R`
`Rightarrow f'(-2)=f'(2)="for all "x in R`