Question

$f\left(x\right)$ is a function defined on entire number line and is even and odd at the same time. Find the value of $f\left(10\right)\times f\left(5\right)$.

___

Answer 1

Given that f(x) is both even and odd function.

f (x) = f (-x) [ f(x) is an even function ] -----(1)

f (x) = - f (-x) [ f(x) is an odd function ] -----(2)
Using (1) and (2) we get,
$\Rightarrow$ f(-x)= - f (-x)

$\Rightarrow$ 2 f(-x) = 0

$\Rightarrow$ f(-x) = 0

$\Rightarrow$ f(x)= 0

f(10) = 0

f(5) = 0

Therefore $f\left(10\right)\times f\left(5\right)$ = 0