Let $f$ and $g$ be continuous functions such that $f(x) ≤ g(x)$ for all $x ∈ [0, 1]$. Determine which of the following statements are true and which are false:
$$ \begin{align} (a) & {}\quad \int_0^x |f(t)|~dt \leq\int_0^x |g(t)|~dt ~\forall~x ∈ [0, 1]\\ (b) & {}\quad \int_0^x (|f(t)|+f(t))~dt \leq\int_0^x \left(|g(t)|+g(t)\right)~dt ~\forall~x ∈ [0, 1]\\ (c) & {}\quad \int_0^x (|f(t)|-f(t))~dt \leq\int_0^x \left(|g(t)|-g(t)\right)~dt ~\forall~x ∈ [0, 1] \end{align}$$ For any statement which you believe to be true, you need to give a proof and for any statement which you believe to be false, you need to give a counter example.
Where to start and how I proceed? Please help me.