Question

If f and g are continuous functions on [a,b] with integral from a to x f ≥ integral from a to x g for every x ∈ [a, b], must it be true that f(x) ≥ g(x) on [a, b]?

Answer 1

$\left[ a,b \right] =\left[ 0,1 \right] \\f\left( x \right) =1-x,g\left( x \right) =x\\\int_a^x{f\left( t \right) dt}=\int_0^x{\left( 1-t \right) dt}=x-\frac{x^2}{2}\\\int_a^x{g\left( t \right) dt}=\int_0^x{tdt}=\frac{x^2}{2}\\x\in \left[ 0,1 \right] \Rightarrow x\geqslant x^2\Rightarrow x-\frac{x^2}{2}\geqslant \frac{x^2}{2}\Rightarrow \int_a^x{f\left( t \right) dt}\geqslant \int_a^x{g\left( t \right) dt}\\f\left( \frac{3}{4} \right) =\frac{1}{4}<\frac{3}{4}=g\left( \frac{3}{4} \right) \\The\,\,statement\,\,is\,\,false.$