# let f(x+y/2)= f(x)+ f(y) / 2 for all real x and y . if f'(0) exists and equals -1 and f(0)=1,find f(2)

Arun Kumar IIT Delhi
8 years ago
$f(0+)'=((f(0)+f(2h))/2-f(0))h=(f(2h)-1)/2h \\=>-1=(f(2h)-1)/2h$

Denominator is going to zero but limit is constant means

$1 \to lim_{2h \to 0}f(2h)=1\\ 2 \to 2f'(2h)=-2\\ =>f'(h)=-x+c \\=>c=+1$

f(2)=1

Thanks & Regards
Arun Kumar
IIT Delhi