Question

Let $f\left(x\right)$ be a continuous function such that $f\left(x\right)$does not vanish for all $xϵR.$ If ${\int }_2^{3}f\left(x\right)dx={\int }_{-2}^3$ $f\left(x\right)dx$ then $xϵR,$ is

• A. an even function
• B. an odd function
• C. a periodic function
• D. none of these

Answer 1

The correct option is B an odd function

Type of function $f\left(x\right)$

${\int }_2^{3}f\left(x\right)dx={\int }_{-2}^{3}f\left(x\right)dx{\int }_2^{3}f\left(x\right)dx={\int }_{-2}^{2}f\left(x\right)dx+{\int }_2^{3}f\left(x\right)dx$

Given that $f\left(x\right)$ does not vanishes for all $x+R$

$f\left(x\right)$ is not equal to zero for any $x$

${\int }_{-2}^{2}f\left(x\right)dx=0$

$f\left(x\right)$ is an odd function

Hence $f\left(x\right)$ is an odd function