Question

Let $f:R\to R$ is a function satisfying $f(2-x)=f(2+x)$ and $f(20-x)=f\left(x\right),\forall x\in R$. For this function $f$ answer the following question.If $f\left(2\right)\ne f\left(6\right)$, then

• A. Fundamental period of $f\left(x\right)$ is 16
• B. Fundamental period of $f\left(x\right)$ may be 12
• C. Period of $f\left(x\right)$ can't be 12
• D. Fundamental period of $f\left(x\right)$ is 8

Answer 1

Answer: (A)

Fundamental period of $f\left(x\right)$ is 16

$f(2-x)=f(2+x)$ ........ $\left(i\right)$
and $f(20-x)=f\left(x\right)$ ...... $\left(ii\right),\forall xϵR$

Replacing $x$ by $2-x$ in equation $\left(i\right)$
$f\left(x\right)=f(4-x)$ ........ $\left(iii\right)$

By comparing $\left(ii\right)$ & $\left(iii\right)$

$f(20-x)=f(4-x)$

$replaceby4-x$

$\Rightarrow f\left(x\right)=f(x+16)$

Hence fundamental period of $f\left(x\right)$ is 16.

Therefore, $\left(A\right)$ is the correct ans