Let $f$ be a bounded function with domain the interval $[0, 1]$. Assume we have found two
partitions $P$ and $Q$ of $[0, 1]$ whose lower and upper sums satisfy
$L_f(Q) = 2$, $U_f(Q) = 5$,
$L_f(P) = 3$, $U_f(P) = 6$.
For each of the following statements, decide whether it is necessarily true (T), it is necessarily
false (F), or it cannot be determined (N). Circle the correct answers on Page 2.
(a) There exists a partition $R$ of $[0, 1]$ such that $L_f(R) \ge 3$ and $U_f(R) \le 5$.
(b) Every partition $R$ of $[0, 1]$ satisfies that $L_f(R) \ge 3$ and $U_f(R) \le 5$.
(c) The lower integral satisfies $\underline{\int_0^1} f \ge 3$.
(d) The lower integral satisfies $\underline{\int_0^1} f \le 5$.
(e) $f$ is integrable on $[0, 1]$.
