Question

If $f(3x+2)+f(3x+29)=0\forall x\in R$, then the period of $f\left(x\right)$ is

• A. $7$
• B. $8$
• C. $10$
• D. none of these

Answer 1

The correct option is D none of these

$f(3x+2)+f(3x+29)=0$
Replace $3x$ by $x$
$\Rightarrow f(x+2)+f(x+29)=0\Rightarrow \left(1\right)$
Replace $x$ by $x+27$
$\Rightarrow f(x+29)+f(x+56)=0\Rightarrow \left(2\right)$
Thus from (1) and (2)
$f(x+2)=f(x+56)$
Replace $x$ by $x-2$
$\Rightarrow f\left(x\right)=f(x+54)$
Hence period of f is $54$