Mathematics - Application of Derivative Question with Solution | TestHub

$F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right).dt$
$F^{\text{'}}\left(x\right)=f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-5\right)$

$\Rightarrow F\left(x\right)$has maxima at$x=2$
and minima at$x=1$and$x=5$
now$F\left(2\right)=\underset{0}{\overset{2}{\int }}\left(x^3-8x^2+17x-10\right).dx$
$={\left[\frac{x^4}{4}-\frac{8x^3}{3}+\frac{17x^2}{2}-10x\right]}_0^2$
$F\left(2\right)=\frac{-10}{3}$which is a maxima in$x\in \left(0, 5\right)$
$F\left(0\right)=0$

Hence$F\left(x\right)<0\forall x\in \left(0, 5\right)$