Let `f((x+y)/(2))= (f(x)+f(y))/(2)` for all real `x and y`. If `f'(0)` exits and equals -1 and `f(0) =1`, then find `f(2)`.

Options

A. `-1`
B. 1
C. 0
D. none of these

Answer

A

Explanation

We have
`f((x+y)/(2))=(f(x)+f(y))/(2)"for all "x,y in R..........(i)`
`Rightarrow f((x+0)/(2))=(f(x)+f(0))/(2)"for all x"in R" "["Putting y=0"]`
`Rightarrow ((x)/(2))=(f(x)+1)/(2)"for all x"in R`
`Rightarrow f(x)=2f((x)/(2))-1"for all x"in R..........(ii)`
Now
`f'(2)=underset(h to 2)lim (f(2+h)-f(2))/(h)`
`Rightarrow f'(2)=underset(h to )lim (f((4+2h)/(2))-f(2))/(h)`
`Rightarrow f'(2)=underset(h to )lim ((f(4)f(2h)-f(2))/(2)-f(2))/(h)" "["Using (i)"]`
`Rightarrow f'(2)=underset(h to )lim (f(4)+f(2h)-2f(2))/(2h)`
`Rightarrow f'(2)=underset(h to )lim (2f(2)-1+2f(h)-1-2f(2))/(2h)["Puttingx=4 and 2h respectively in (ii)"]`
`Rightarrow f'(2)=underset(h to )lim (f(h)-1)/(h)`
`Rightarrow f'(2)=underset(h to )lim (f(h)-f(0))/(h)=f'(0)=-1`