Let f :[(1)/(2)1] ⟶ R (the set of all real numbers) be a positive,...

${f}^{'}\left(x\right) < 2f\left(x\right)$
$\text{⇒}\dfrac{{f}^{'}\left(x\right)}{f\left(x\right)} < 2$

$⇒ {∫}_{\dfrac{1}{2}}^{x}\dfrac{f ' \left(x\right)}{f\left(x\right)}dx = {∫}_{\dfrac{1}{2}}^{x}2dx$

$⇒ lnf\left(x\right) < 2x - 1$

$⇒ f\left(x\right) < {e}^{2x - 1}$
$⇒ 0 < {∫}_{\dfrac{1}{2}}^{1}f\left(x\right)dx < {∫}_{\dfrac{1}{2}}^{1}{e}^{\left(2x - 1\right)}dx$

$⇒ 0 < {∫}_{\dfrac{1}{2}}^{1}f\left(x\right)dx < {\left[\dfrac{{e}^{2x - 1}}{2}\right]}_{\dfrac{1}{2}}^{1}$
$⇒ 0 < {∫}_{\dfrac{1}{2}}^{1}f\left(x\right)dx < \dfrac{e - 1}{2}$