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 (0) $\ne$ 0 then-

• A. is an even function
• B.
• C.
• D.

Answer 1

Answer: (A), (B)

The correct options are
A

is an even function

B

2 f (x) f(y) = f (x - y) + f ( x + y) at x = y = 0 ; f (0) = 1 ( f (0) $\ne$ 0 )

and at y = 0

2 f (x) = f (- x) + f (x) $\Rightarrow$ f (x) = f ( - x)

ie . f (x) is an even function $f^{{}^{\prime }}$(x) = - $f^{{}^{\prime }}$ ( - x)

$\Rightarrow$$f^{{}^{\prime }}$ (1) + f (-1) = 0

Also $f^{{}^{\prime }}$(x) = - $f^{{}^{\prime }}$ (- x)

$f^{{}^{\prime }}$(x) + $f^{{}^{\prime }}$ (- x) = 0