Let $f$ be a continuous function [a, b] such that 

$f\left(x\right)>0\text{ for all }x\in [a,b]\text{. If }F\left(x\right)={\int }_a^x f\left(t\right)\mathrm{d}t$ then

$F$ is differentiable but not increasing on [a, b]

$F$ is differentiable and increasing on [a, b]

$F$ is continuous and decreasing on [a, b]

$F$ is neither differentiable nor increasing on [a, b]