Question

Function f(x) is positive, decreasing and concave up on the closed interval
[a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the left sum, right sum, and trapezoidal rule approximations for the value of integral from a to b f(x)dx. Which one of the following statements is true?
a) Left sum < trapezoidal rule value < Right sum
b) Left sum < Right sum < trapezoidal rule value
c) Right sum <trapezoidal rule value < left sum
d) Cannot be determined without the x-values for the partitions

Answer 1

just draw a graph, such as y = 1/x, which fits the criteria.
Make it big enough that you can draw the desired rectangles and trapezoids.

It should then be easy to see which ones over- and under-estimate, and then decide which order they belong in.

Answer 2

I don't get it. Please help me!

Answer 3

a) Left sum < trapezoidal rule value < Right sum