If f(x - y), f(x) f(y) and f(x + y) are in A.P. for all x, y ∈ R and f(0) = 0, then
Answer & explanation
Correct answer: option 1
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.
∴ $2 f(x) f(y)=f(x-y)+f(x+y)$ for all $x, y \in R$
$\Rightarrow 2 f(0) f(x)=f(-x)+f(x)$ for all $x \in R$ [Replacing x by 0 and y by x]
$\Rightarrow f(-x)+f(x)=0$ 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 $x \in R$
$\Rightarrow f'(-2)=f'(2)$