Question

If f(x – y) ,f(x) f(y) and f(x + y) are in A.P for all $x,y\in R$ and $f\left(0\right)\ne 0$ then

• A. f (x) is an even function
• B. f'(1) + f'(–1) = 0
• C. f'(2) – f'(–2) = 0
• D. f(3) + f(–3) = 0

Answer 1

Answer: (A), (B)

The correct options are
A f (x) is an even function
B f'(1) + f'(–1) = 0

2f(x)f(y) = f(x - y) + f (x + y) at x= y = 0;f(0)=1 $(\because f(0)\ne 0)$
and at y = 0
2f(x) = f (–x) + f(x) $\Rightarrow f\left(x\right)=f(-x)$
ie. f(x) is an even function f'(x) = – f'(–x)
$\Rightarrow f^{\prime }\left(1\right)+f(-1)=0$
Also f'(x) = – f'(-x)
f'(2) + f'(-2) = 0