Prove/disprove the following:

Question

I'm given the following problem:

Prove that if $f$ is negative and increasing between $a$ and $b$, then the left Riemann sum is an underestimate and the right Riemann sum is an overestimate.

After drawing a graph, I concluded that this statement is false but I do not know how to prove this algebraically. Any pointers?

Answer 1

HINT:

Consider the partition $[x_0, x_1, x_2, \cdots, x_n]$ of $[a, b]$ where $x_0 = a, x_n = b$;

Notice that $f(x_0) \leq f(x_1) \leq \cdots \leq f(x_{n-1}) \leq f(x_n)$. (the $\leq$ may be exchanged by $<$ if the problem statement means that $f$ is strictly increasing.

Are you able to write the expressions for the left and right riemman sums of that partition?