Question

If $f\left(x\right)$ is a continuous function such that $f\left(x\right)\ge 0,\forall x\in [2,10]$ and ${\int }_4^{8}f\left(x\right)dx=0$, then the value of $f\left(6\right)$, is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is B $0$

$f\left(x\right)$ is above the x-axis or on the x-axis for all $x\in [2,10]$.
If $f\left(x\right)$ is greater than zero at any sub-interval of $[4,8]$, then ${\int }_4^{8}f\left(x\right)dx$ must be greater than zero.
But ${\int }_4^{8}f\left(x\right)dx=0$, which shows $f\left(x\right)$ can't have any value greater than zero in the sub-interval $[4,8]$.
$⟹f\left(x\right)$ is constant in the sub-interval $[4,8]$ and has to be zero at all points, $x\in [4,8]$.
$⟹f\left(x\right)=0,\forall x\in [4,8]⟹f\left(6\right)=0$.
Ans: B