Question

A function $f:R\to R$ satisfying the following conditions
$i\left)f\right(-x)=f(x)$
$ii\left)f\right(x+2)=f(x)$
$iiig\left(x\right)={\int }_0^{x}f\left(t\right)dt$ and
$g\left(1\right)=\alpha$ then
The value of $g\left(2\right)$ and $g\left(5\right)$ in terms of $\alpha$ is

• A. $2\alpha ,5\alpha$
• B. $\alpha ,\alpha$
• C. $\alpha /2,\alpha /5$
• D. none of these

Answer 1

The correct option is A $2\alpha ,5\alpha$

$f(-x)=f\left(x\right)⟶$Even function

$f(x+2)=f\left(x\right)$

$g\left(x\right)={\int }_0^{x}f\left(t\right)dt\Rightarrow g\left(1\right)={\int }_0^{1}f\left(t\right)dt=\alpha \Rightarrow g\left(2\right)={\int }_0^{2}f\left(t\right)dt=2{\int }_0^{1}f\left(t\right)dt=2\alpha \Rightarrow g\left(2\right)=2\alpha$

$g\left(5\right)={\int }_0^{5}f\left(t\right)dt={\int }_0^{2}f\left(t\right)dt={\int }_2^{4}f\left(t\right)dt={\int }_4^{5}f\left(t\right)dt\Rightarrow 2\alpha +2\alpha +\alpha \Rightarrow 5\alpha \Rightarrow g\left(5\right)=5\alpha$