Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let
\[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\]
Which of the following is $\text{TRUE}?$
- There exists an $f$ so that
\[A+B<\int_{0}^{2} f(x) d x\]
- There exists a strictly negative $f$, that is $f(x)<0$ for all $x \in[0,2]$, such that
\[\int_{0}^{2} f(x) d x=A+B=B-A\]
- There exists such an $f$ so that
\[\int_{0}^{2} f(x) d x=A+B=A-B\]
- There does not exist an $f$ such that $\int_{0}^{1} f(x) d x=3$
- There does not exist an $f$ so that
\[A+B \leq-\int_{0}^{2} f(x) d x\]