Question

Let $f\left(x\right)$ and $\varphi \left(x\right)$ are two continuous functions on $R$ satisfying $\varphi \left(x\right)=\underset{a}{\overset{x}{\int }}f\left(t\right)dt,a\ne 0$ and another continuous function $g\left(x\right)$ satisfying $g(x+\alpha )+g\left(x\right)=0\forall x\in R,\alpha >0$, and $\underset{b}{\overset{2k}{\int }}g\left(t\right)dt$ is independent of $b$.

Least positive value of $c$ if $c,k,b$ are in $A.P.$ is

• A. $0$
• B. $1$
• C. $\alpha$
• D. $2\alpha$

Answer 1

The correct option is D $2\alpha$

$g(x+\alpha )+g\left(x\right)=0\Rightarrow g(x+2\alpha )+g(x+\alpha )=0\Rightarrow g(x+2\alpha )=g\left(x\right)$

Thus, $g\left(x\right)$ is periodic with period $2\alpha .$
$\therefore \underset{b}{\overset{2k}{\int }}g\left(t\right)dt=\underset{b}{\overset{b+c}{\int }}g\left(x\right)dx[\because b,k,c\text{are in}A.P.]$

This is independent of $b.$ Then $c$ has least value of $2\alpha$.