Help needed thank you!

Question

Function f(x) is positive, increasing 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 upper sum, lower sum, and trapezoidal rule approximations for the value of .

Which one of the following statements is true?

Options:

1)	Lower sum < Trapezoidal rule Value < Upper sum
2)	Lower sum < Upper sum < Trapezoidal rule value
3)	Trapezoidal rule < Lower sum < Upper sum
4)	Cannot be determined without the x-values for the partitions

Answer 1

Good Question!

In this problem, the first option is true. This is because when a function is increasing, an upper sum will overestimate the area, while a lower sum will underestimate it.

Since the function is concave up, the trapezoidal rule sum will also underestimate the area, but not so drastically as the lower sum.