Question

A function $f:R\to R$ satisfies the following conditions:
1) $f(-x)=f\left(x\right)$
2) $f(x+2)=f\left(x\right)$
3) $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$ and $g\left(1\right)=\alpha$
Then, the value of $g(x+2)-g\left(2\right)$ is

• A. $3g\left(x\right)$
• B. $2g\left(x\right)$
• C. $g\left(x\right)$
• D. none of these

Answer 1

Answer: (C)

$g\left(x\right)$

$f\left(x\right)=f(-x)$ ...(1)
$f(x+2)=f\left(x\right)$ ...(2)
From (1) and (2), we can say that $f\left(x\right)$ can be a constant function.
$\therefore f\left(x\right)=k$
$g\left(x\right)={\int }_0^{x}f\left(x\right)dx={\int }_0^{x}kdx=kx$
As given, $g\left(1\right)=\alpha \Rightarrow k=\alpha$
$\therefore g\left(x\right)=x\alpha$
Now,
$g(x+2)-g\left(2\right)=\alpha (x+2)-2\alpha =x\alpha =g\left(x\right)$
Hence, option C is correct.