Suppose that a < b. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples for the false ones.
a) If f is bounded on [a, b], if g is absolutely integrable on (a, b), and if |(x)| < g(x) for all x ∈ (a, b), then f is absolutely integrable (a, b).
b) Suppose that h is absolutely integrable on (a, b). If f is continuous (a, b), if g is continuous and never zero on [a, b], and if |f(x) < h(x) for all x ∈[a, b], then f/g is absolutely integrable on (a, b).
c) If f : (a, b) → [0, ∞) is continuous and absolutely integrable (a, b) for some a, b, then √f is absolutely integrable on (a, b).
d) If f and g are absolutely integrable on (a, b), then max{f, g) min{f, g} are both absolutely integrable on (a, b).