Discuss the validity of each statement. If the statement is always true, explain why. If it is not always true, give a counterexample. If the area under the graph of f on [a, b] is equal to both the left sum Ln​ and the right sum Rn​ for some positive integer n, then f is constant on [a, b].
A. The statement is always true because the only time the left sum equals the right sum is for constant functions.
B. The statement is false. The area under the graph of f(x) = ∣x − 1∣ on [0, 2] satisfies the given conditions, but is not constant.
C. The statement is always true because if a function is either increasing or decreasing over an interval, the left sum will not equal the right sum.
D. The statement is false. Any linear function on a closed and bounded interval satisfies the given conditions, but is not necessarily constant.