Question

Consider $f:R\to R,y=f\left(x\right)$ be a function which is symmetrical with respect to two vertical lines $x=4$ and $x=10$ then

• A. $f\left(x\right)$ must be a constant function
• B. $f\left(x\right)$ must be a periodic function
• C. $f\left(x\right)=6\forall x\in R$
• D. $f\left(x\right)$ will be periodic with period $12$

Answer 1

Answer: (B), (D)

$f\left(x\right)$ will be periodic with period $12$
$f\left(x\right)$ must be a periodic function

If function $f\left(x\right)$ is symmetrical about $x=4$ and $x=10$
$\Rightarrow f(4-x)=f(4+x)....\left(1\right)f(10-x)=f(10+x)....\left(2\right)$

Substitute $x=4-x$ in $\left(1\right)$

$\Rightarrow f\left(x\right)=f(8-x)....\left(3\right)$

Now, substitute $x=x+2$ in $\left(2\right)$

$\Rightarrow f(8-x)=f(12+x)....\left(4\right)$

From $\left(3\right)$ and $\left(4\right)$, we get

$f\left(x\right)=f(12+x)$
$\Rightarrow f\left(x\right)$ is periodic with period 12.