Let f be a continuous function [a, b] such that
f(x)>0 for all x∈[a,b]. If F(x)=∫ax f(t)dt 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]
F′(x)=f(x)>0
⇒ F is differentiable and increasing on [a, b].